ABSTRACT EVIDENCE FOR INTRINSIC REFLECTION … Lab Theses 1965... · ABSTRACT NUCLEAR SHAPES Paul Cottle Yale University 1986 Most of the important advances in the description of

ABSTRACT

NUCLEAR SHAPES

Paul Cottle

Yale University

1986

Most of the important advances in the description of collective

nuclear phenomena have been based upon the assumption of a reflection

symmetry relative to a plane perpendicular to the symmetry axis of the

nucleus. It is known, however, that reflection asymmetric shapes, such

as alpha clustering configurations and dynamic octupole deformations,

can be found in many regions of the periodic table.

Isotopes of Ra and Th have long been known to exhibit characteris

tics suggesting the possibility that static intrinsic reflection asymme

tric shapes are present. We examine the nature of reflection asymmetry

in Ra and Th by first studying, in detail, three nuclei, 216Rn, 219Ra

and 220Ra, produced in the 14C+208Pb fusion reaction using several tech

niques of gamma ray spectroscopy. Level spectra have been constructed

for these nuclei to excitation energies of 2.1 MeV, 4.9 MeV and 3.6 MeV,

respectively. Second, we discuss our results within the context of the

systematic behavior of other Rn, Ra and Th isotopes and its relationship

to predictions of alpha particle cluster, octupole vibration and static

octupole deformation models. Third, we find that several observables

seen in the Ra-Th-region are qualitatively similar to those found in the

EVIDENCE FOR INTRINSIC REFLECTION ASYMMETRIC

Sm-Gd region. Finally, we see that the excitation spectra observed in

the mass 217 and 219 Ac and Ra isotopes are described quite well in

terms of a weak particle-core coupling model which is generally success

ful for weakly deformed nuclei.

We have found clear evidence for further reflection asymmetric

shapes in the light actinide region. We conclude, however, that because

of the similarities of the above models we are at present unable to dis

tinguish which model is most appropriate for the description of these

nuclei. In addition, we find that the weak coupling model is appropri

ate for the description of odd-A nuclei in the weakly deformed Ra-Ac

region.

NUCLEAR SHAPES

EVIDENCE FOR INTRINSIC REFLECTION ASYMMETRIC

A Dissertation

Presented to the Faculty of the Graduate School

of

Yale University

in Candidacy for the Degree of

Doctor of Philosophy

by

Paul Davis Cottle

June 1986

To Mom, Dad and Tracey

ACKNOWLEDGMENTS

I have been fortunate enough to have had the support of many out

standing people over the last four years. Unfortunately, I can only

mention some of them here.

I must first thank Prof. D. Allan Bromley, who contributed in many

ways to this work. His endless encouragement, which meant so much com

ing from a man of his stature, was indispensible in both the research

leading to this dissertation and the process of making decisions about

my future. Prof. Bromley has my eternal gratitude.

Prof. Moshe Gai's wide knowledge of the field of nuclear physics

has served as an inspiration to me as well as a valuable resource for my

own learning. The contents of this dissertation strongly reflect his

"big picture" approach to nuclear physics. ■

I also wish to acknowledge the contributions that Prof. Jolie Ciz-

ewski has made to my education and to my self-confidence. I hope that

Jolie1s influence on my work, in the future as well as the present, will

be obvious to anyone familiar with her accomplishments.I

Prof. John Shriner has also played an important role in both my

life and my short career. I owe John a great debt for his friendship

and help. ,

My fellow graduate students made working in this Laboratory a plea

sant and rewarding experience. I have been fortunate to share a hallway

with John Ennis, Mario Ruscev, Mark Drigert, Steve Sterbenz, Anna Hayes,

Patty Blauner, Laurie Baumel, Kevin Hubbard, Mike Kaurin, John O'Connor,

Niki Becker, Bronek Dichter, Steve Rugari, Zhiping Zhao, Paul Magnus,

Kooverji Gamadia and (how could I ever forget) Tzu-Fang Wang. Prof.

Robert Weller, Dr. Mari Weller, Dr. Nicholas Tsoupas and Prof. Peter

Parker have also helped to make this lab a good place to work.

I was fortunate to have worked with outstanding collaborators from

Brookhaven National Laboratory and SUNY-Stony Brook: Dr. John Olness,

Dr. Ernest Warburton, Con Beausang, Dr. Mubina Quader, Dr. Lars Hild-

ingsson. Dr. William Piel, Jr., and Prof. David Fossan. I also wish to

thank as a group the very helpful people who make up the accelerator

operations and support staffs at Brookhaven.

Finally, I would like to thank all of. those people who have kept

our Laboratory running so smoothly over the past few years, especially

Kenzo Sato, Phil Clarkin, Dick D'Alexander, Teddy Duda, Ed Stepensky,

Tom Barker, Richard Wagner, Charles Gingell, Al Jeddry, Joe Cimino, Ray

Comeau, Nitza Hauser, Rita Bonito, Clare Buckley, Annalee Jacunski,

Karen DeFelice, Marvin Curtis, Sandy Sicignano and Mary Anne Schultz.

Without these people, the rest of us would have been almost completely

lost.

/

TABLE OF CONTENTS

T A B L E O F C O N T E N T S .................................................................................................... V

T A B L E O F F I G U R E S ....................................................................................................... v i i i

T A B L E O F T A B L E S .......................................................................................................... x i

1 . I N T R O D U C T I O N ....................................................................................................... 1

2 . O V E R V I E W O F T H E O R Y ........................................................................................ 2 3

2 . 1 A L P H A P A R T I C L E C L U S T E R I N G I N H E A V Y N U C L E I A N D

T H E H Y B R I D M O D E L ............................................................................. 2 3

2 . 2 O C T U P O L E V I B R A T I O N A L B E H A V I O R I N E V E N - E V E N

N U C L E I ....................................................................................................... 3 3

2 . 3 S T A T I C O C T U P O L E D E F O R M A T I O N S I N E V E N - E V E N

N U C L E I ....................................................................................................... 3 9

2 . 4 A F E W C O M M E N T S O N T H E R E L A T I O N S H I P B E T W E E N T H E

A L P H A C L U S T E R A N D S T A T I C O C T U P O L E M O D E L S ................ 4 6

2 . 5 W E A K A N D I N T E R M E D I A T E C O U P L I N G I N T R A N S I T I O N A L

O D D - A N U C L E I ....................................................................................... 4 7

2 . 6 S T R O N G C O U P L I N G I N O D D - A N U C L E I A N D T H E O C T U P O L E

N I L S S O N M O D E L ..................................................................................... 5 2

2 . 7 P A R I T Y D O U B L E T S I N R E F L E C T I O N A S Y M M E T R I C O D D - A

N U C L E I ...................................................................................................... 5 5

3 . E X P E R I M E N T A L P R O C E D U R E ............................ ................................................ 5 7

3 . 1 G E N E R A L F E A T U R E S ............................................................................. 5 7

ACKNOWLEDGEMENTS ..................................... iii

3 . 2 H E A V Y I O N F U S I O N - E V A P O R A T I O N R E A C T I O N S .................... 5 9

3 . 3 E X P E R I M E N T A L D E T A I L S .................................................................. 6 3

4 . A N A L Y S I S A N D R E S U L T S .................................................................................... 7 1

4 . 1 P R E V I O U S W O R K O N 2 1 6 R n , 2 2 ° R a A N D

2 1 9 R a .. .......................................................................................................... 7 1

4 . 2 E X C I T A T I O N F U N C T I O N M E A S U R E M E N T S ................................... 7 1

4 . 3 2 2 ° R a L E V E L S P E C T R U M .................................................................. 7 3

4 . 4 2 1 6 R n L E V E L S P E C T R U M .................................................................. 7 9

4 . 5 2 1 9 R a L E V E L S P E C T R U M .................................................................. 8 9

5 . D I S C U S S I O N ............................................................................................................. 1 0 0

5 . 1 S Y S T E M A T I C B E H A V I O R O F E V E N R n , R a a n d T h

N U C L E I ....................................................................................................... 1 0 1

5 . 2 I N T E R P R E T A T I O N O F R a A N D T h I S O T O P E S I N T E R M S O F

A N A L P H A P A R T I C L E C L U S T E R M O D E L ....................................... 1 1 4

5 . 3 S T U D Y O F E V E N R a I S O T O P E S T H R O U G H C A L C U L A T I O N S

W I T H T H E U ( 6 ) ® U ( 4 ) H Y B R I D M O D E L ....................................... 1 1 6

5 . 4 P R E D I C T I O N S F O R T H E O C T U P O L E D E G R E E O F F R E E D O M

I N 2 2 ° R a .................................................................................................. 1 2 2

5 . 5 A C O M P A R I S O N O F T H E Z = 6 0 - 6 6 I S O T O P E S W I T H T H E

Z = 8 8 - 9 0 O N E S ....................................................................................... 1 2 3

5 . 6 A C O M P A R I S O N O F I - A N D 3 ‘ S T A T E S I N T H E

R E G I O N S 5 6 < Z < 8 2 A N D Z > 8 2 ......................................................... 1 4 1

5 . 7 S Y S T E M A T I C B E H A V I O R O F O D D - A N U C L E I N E A R A = 2 1 9 . 1 5 6

6 . S U M M A R Y A N D C O N C L U S I O N S ............................................................................... 1 6 4

B I B L I O G R A P H Y ................................................................................................................. 1 6 7

4

8

10

12

1 4

1 7

1 8

2 5

2 7

3 1

3 4

3 5

3 8

4 1

4 3

4 4

4 5

5 1

5 4

5 6

6 1

6 5

P r o p o s e d M o l e c u l a r B a n d i n 1 8 0 .......................

R e d u c e d A l p h a P a r t i c l e D e c a y W i d t h s ..........

A l p h a P a r t i c l e C l u s t e r a n d O c t u p o l e

C o n f i g u r a t i o n s ................................................................

P a r t i a l L e v e l S p e c t r u m o f 1 5 0 G d .....................

2 3 0 U A l p h a P a r t i c l e D e c a y C h a i n .....................

P a r t i a l L e v e l S p e c t r u m o f 2 1 8 R a .....................

P a r t i a l L e v e l S p e c t r u m o f 2 0 N e .......................

C l a s s i c a l U ( 6 ) a n d U ( 4 ) R e p r e s e n t a t i o n s .

S p e c t r o s c o p y o f D y n a m i c a l L i m i t s o f U ( 4 )

S c h e m a t i c H y b r i d M o d e l E x a m p l e .......................

P a r t i a l L e v e l S p e c t r u m o f 1 6 2 E r .....................

B ( E 1 ) / B ( E 2 ) v s . J f o r L a n t h a n i d e N u c l e i .

M o l l e r a n d N i x M a s s C o m p a r i s o n .......................

G r o u n d S t a t e P o t e n t i a l E n e r g y S u r f a c e s . .

S p e c t r o s c o p y o f O c t u p o l e C o n f i g u r a t i o n s .

B a c k b e n d i n g P l o t f o r 2 2 2 T h .................................

W e a k C o u p l i n g M u l t i p l e t s .......................................

W o o d s - S a x o n S i n g l e P a r t i c l e L e v e l s .............

L e v e l S p e c t r u m o f 2 2 5 A c .........................................

l 8 i x a ( 1 6 0 , x n ) E x c i t a t i o n F u n c t i o n ................

TABLE OF FIGURES

Collective Nuclear Motion.............

Compton Suppression Electronics.......

66

6 7

7 2

7 4

7 5

7 6

8 4

8 5

8 7

9 0

9 1

9 2

9 3

102

1 0 4

1 0 6

1 0 7

112

1 1 3

1 1 5

1 1 7

1 1 9

120

G a m m a - g a m m a T i m i n g E l e c t r o n i c s ....................................

G a m m a - g a m m a E n e r g y E l e c t r o n i c s ....................................

i 4 c + 2 0 8 p b E x c i t a t i o n F u n c t i o n .......................................

L e v e l S p e c t r u m o f 2 2 0 R a ......................................................

2 2 0 R a G a m m a - g a m m a G a t e s ......................................................

2 2 0 R a A n g u l a r D i s t r i b u t i o n S p e c t r u m .......................

2 1 6 R n G a m m a - X - r a y G a t e ........................................................

2 1 6 R n G a m m a - g a m m a G a t e ........................................................

L e v e l S p e c t r u m o f 2 1 6 R n ......................................................

2 1 9 R a G a m m a - g a m m a G a t e s ......................................................

2 i g R a A n g u l a r D i s t r i b u t i o n S p e c t r u m .......................

L e v e l S p e c t r u m o f 2 1 9 R a ......................................................

G r o u n d S t a t e S p i n s o f E v e n - Z , O d d - N Z = 8 2 - 9 2

N u c l e i ..................................................................... ...........................

S y s t e m a t i c B e h a v i o r o f R a E n e r g y L e v e l s .............

I ( x ) v s . f iw f o r 2 1 8 R a , 2 2 0 R a .........................................

S y s t e m a t i c B e h a v i o r o f T h E n e r g y L e v e l s .............

S y s t e m a t i c B e h a v i o r o f E n e r g y L e v e l s i n N = 1 3 0

I s o t o n e s a n d R n I s o t o p e s ...................................................

R a B ( E 1 ) / B ( E 2 ) R a t i o s ...........................................................

T h B ( E 1 ) / B ( E 2 ) R a t i o s ...........................................................

A l p h a P a r t i c l e H i n d r a n c e F a c t o r s f o r Z = 8 6 - 9 4 .

E v i d e n c e f o r A l p h a P a r t i c l e C l u s t e r i n g ...............

E p i n H y b r i d M o d e l C a l c u l a t i o n ....................................

Hybrid Model Calculation Results...........

5 . 1 1 P a r t i a l L e v e l S p e c t r u m o f 1 4 4 S m ............................................ 1 2 5





5 . 1 6 P a r t i a l L e v e l S p e c t r u m o f 1 4 6 G d ............................................ 1 3 0




5 . 2 0 S y s t e m a t i c B e h a v i o r o f S m E n e r g y L e v e l s ........................ 1 3 4

5 . 2 1 S y s t e m a t i c B e h a v i o r o f G d E n e r g y L e v e l s ........................ 1 3 5

5 . 2 2 S y s t e m a t i c B e h a v i o r o f R a E n e r g y L e v e l s ........................ 1 3 6

5 . 2 3 G r o u n d S t a t e A l p h a P a r t i c l e D e c a y W i d t h s f o r

L a n t h a n i d e a n d A c t i n i d e N u c l e i ............................................... 1 3 7

5 . 2 4 B ( E 1 ) / B ( E 2 ) V a l u e s f o r L a n t h a n i d e a n d A c t i n i d e

N u c l e i ............................................................................................................ 1 3 8

5 . 2 5 G l o b a l R a n g e o f V a l u e s f o r B ( E 1 ) / B ( E 2 ) ........................... 1 3 9

5 . 2 6 E ( 3 " ) v s . N f o r L a n t h a n i d e a n d A c t i n i d e N u c l e i . 1 4 31 n

5 . 2 7 N i l s s o n D i a g r a m f o r Z = 5 0 - 8 2 ....................................................... 1 4 5

5 . 2 8 N i l s s o n D i a g r a m f o r N = 8 2 - 1 2 6 .................................................... 1 4 6

5 . 2 9 N i l s s o n D i a g r a m f o r Z > 8 2 ............................................................... 1 4 7

5 . 3 0 N i l s s o n D i a g r a m f o r N > 1 2 6 ............................................................ 1 4 8

5 . 3 1 E ( 3 “ ) v s . N f o r L a n t h a n i d e a n d A c t i n i d e N u c l e i . 1 4 91 P

5 . 3 2 E ( 3 _ " ) - E ( l " ) v s . N f o r L a n t h a n i d e a n d A c t i n i d e' 1 1 n

N u c l e i ............................................................................................................ 1 5 3

5.33

5 . 3 4

a n d A c t i n i d e N u c l e i ........................................................................... 1 5 4

P a r t i c l e - C o r e C o u p l i n g f o r A = 2 1 7 a n d 2 1 9 ..................... 1 5 8

E(31-)-E(l1") vs. E(41+)/E(21+) for Lanthanide

TABLE OF TABLES

4 . 1 C h a r a c t e r i s t i c s o f 2 2 0 R a G a m m a R a d i a t i o n 8 0

4 . 2 B ( E 1 ) / B ( E 2 ) V a l u e s f o r 2 2 0 R a ............................................. 8 2

4 . 3 C h a r a c t e r i s t i c s o f 2 1 6 R n G a m m a R a d i a t i o n 8 8

4 . 4 C h a r a c t e r i s t i c s o f 2 1 9 R a G a m m a R a d i a t i o n 9 6

4 . 5 B ( E 1 ) / B ( E 2 ) V a l u e s f o r 2 1 9 R a ............................................. 9 9

5 . 1 P a r a m e t e r s U s e d i n V i b r o n M o d e l F i t s ........................ 1 2 1

1. INTRODUCTION

t a l l y o b s e r v e d n u c l e i m a y b e c l a s s i f i e d a s b e l o n g i n g t o o n e o f t w o c a t

e g o r i e s . M e m b e r s o f t h e f i r s t c a t e g o r y l e n d t h e m s e l v e s t o a s h e l l m o d e l

a n a l y s i s r e m i n i s c e n t o f t h a t a p p l i c a b l e t o e l e c t r o n i c b e h a v i o r i n a t o m s .

I n c o n t r a s t , e a c h n u c l e a r m a n y - b o d y s y s t e m i n t h e s e c o n d c a t e g o r y s e e m s

m o s t n a t u r a l l y t r e a t e d a s a c l a s s i c a l l i q u i d d r o p w h i c h d i s p l a y s v a r i e d

v i b r a t i o n a l a n d s t a t i c a l l y d e f o r m e d r o t a t i o n a l m o d e s . C l e a r l y , h o w e v e r ,

t h i s i s a c a r i c a t u r e ; m o s t n u c l e i d i s p l a y e l e m e n t s o f b o t h t y p e s o f

b e h a v i o r .

E a r l y s t u d i e s o f t h e n a t u r a l i s o t o p i c a b u n d a n c e s o f h e a v y n u c l e i

r e v e a l e d d i s t r i b u t i o n s s t r o n g l y p e a k e d a t n e u t r o n o r p r o t o n n u m b e r s o f

5 0 , 8 2 a n d 1 2 6 . T h e s e " m a g i c " n u m b e r s , a s t h e y w e r e n a m e d , w e r e a l s o

h i g h l i g h t e d b y t h e s t r i k i n g b e h a v i o r e x h i b i t e d b y n e u t r o n s e p a r a t i o n

e n e r g i e s , n e u t r o n c a p t u r e c r o s s s e c t i o n s a n d e l e c t r i c q u a d r u p o l e m o m e n t s

( M a 4 8 ) . M e a s u r e m e n t s o f t h e s e q u a n t i t i e s , a s w e l l a s o t h e r o b s e r v

a b l e s , w e r e e x p l a i n e d q u i t e s u c c e s f u l l y i n t e r m s o f a s p h e r i c a l n u c l e a r

s h e l l m o d e l a s l o n g a s N a n d Z w e r e w i t h i n a f e w n u c l e o n s o f m a g i c n u m

b e r s ( M a 5 5 ) .

H o w e v e r , n u c l e i i n w h i c h N a n d Z w e r e b o t h q u i t e r e m o v e d f r o m t h e

m a g i c n u m b e r s s h o w e d e x c i t a t i o n s p e c t r a w h i c h c o u l d n o t b e r e p r o d u c e d

w i t h s i m p l e s h e l l m o d e l s . T h i s w a s f i r s t n o t i c e d i n t h e s t u d y o f

n u c l e a r e l e c t r i c q u a d r u p o l e m o m e n t s , w h e r e v a l u e s m u c h l a r g e r t h a n t h o s e

p o s s i b l e f o r a s i n g l e v a l e n c e n u c l e o n w e r e o b s e r v e d ( T o 4 9 ) . A n u c l e a r

c o l l e c t i v e p i c t u r e b a s e d o n t h e c o h e r e n t m o t i o n o f m a n y n u c l e o n s a n d t h e

On a simple level, the low energy spectra of the 2500 experimen

r e s e m b l a n c e o f t h i s s y s t e m t o a c l a s s i c a l l i q u i d d r o p w a s b u i l t u p o n

t h i s e x p e r i m e n t a l i n f o r m a t i o n ( R a 5 0 , B o 5 2 , B o 7 5 ) . I n t h e m o s t g e n

e r a l s e n s e , t h e s h a p e o f a l i q u i d d r o p c a n b e w r i t t e n i n t e r m s o f s p h e r

i c a l h a r m o n i c s a s

R = R n { 1 + E a . ( t ) Y (e ,</>) } ( 1 . 1 )0 Jim Urn *

w h e r e t h e a . ( t ) a r e i n g e n e r a l t i m e d e p e n d e n t m e a s u r e s o f t h e v a r i o u s Hm

m u l t i p o l e d e f o r m a t i o n s o f t h e s h a p e a n d R ^ i s t h e m e a n r a d i u s . B e c a u s e

o f t h e r e l a t i v e i n c o m p r e s s i b i l i t y o f n u c l e a r m a t t e r , t h e m o n o p o l e t e r m

i s o f l i t t l e i n t e r e s t i n t h e s t u d y o f l o w e n e r g y n u c l e a r s p e c t r a . F u r

t h e r , t h e d i p o l e t e r m c o r r e s p o n d s t o a s i m p l e t r a n s l a t i o n o f t h e n u c l e u s

w i t h n o c h a n g e i n i n t r i n s i c s h a p e . T h u s , t h e l o w e s t o r d e r c o l l e c t i v e

m o d e , a s w e l l a s t h e d o m i n a n t o n e , i s t h e q u a d r u p o l e m o d e .

W e w i l l r e s t r i c t o u r d i s c u s s i o n t o a x i a l l y s y m m e t r i c s h a p e s , a n d

d e f i n e , i n t h e c o n v e n t i o n a l m a n n e r , a n e w s e t o f s h a p e v a r i a b l e s , 6 , b yw

R = R q ' { 1 + Z £ 3 ^ ( 0 , * ) J ( 1 - 2 )

w h e r e R Q ‘ i s t h e r a d i u s o f a s p h e r e w i t h t h e s a m e v o l u m e a s t h e n u c l e a r

s h a p e .

B e c a u s e o f t h e d o m i n a n c e o f t h e q u a d r u p o l e m o d e t h r o u g h o u t t h e

p e r i o d i c t a b l e a n d t h e r e l a t i v e w e a k n e s s o f o t h e r m u l t i p o l a r i t i e s , m o s t

v e r s i o n s o f t h e c o l l e c t i v e m o d e l d e v e l o p e d i n t h e 3 7 y e a r s s i n c e t h i s

-2-

-3-

p i c t u r e c a m e t o t h e f o r e h a v e d e a l t e x c l u s i v e l y w i t h t h e q u a d r u p o l e

d e g r e e o f f r e e d o m . S m a l l c o n t r i b u t i o n s t o t h e s h a p e f r o m h e x a d e c a p o l e

t e r m s h a v e b e e n i d e n t i f i e d i n t h e l a s t t w e n t y y e a r s ( A p 7 0 , H e 6 8 ) . A s

a n e x a m p l e o f t h e i r r e l a t i v e i m p o r t a n c e , w e c a n . p o i n t o u t t h a t i n t h e

m o s t d e f o r m e d h e a v y n u c l e i a p p r o x i m a t e l y 0 . 3 5 i n t h e g r o u n d s t a t e ,

A n i n t r i n s i c s h a p e c o m p o n e n t c a n b e e i t h e r o s c i l l a t o r y ( d y n a m i c a l l y

d e f o r m e d , o r v i b r a t i o n a l ) o r n o n - o s c i l l a t o r y ( s t a t i c a l l y d e f o r m e d ) . T h e

e n e r g y s p e c t r u m c h a r a c t e r i s t i c o f t h e s i m p l e s t c a s e o f v i b r a t i o n a l

b e h a v i o r , h a r m o n i c o s c i l l a t i o n , s h o w s a s e r i e s o f e q u a l l y s p a c e d a n d

d e g e n e r a t e m u l t i p l e t s . A s t a t i c a l l y d e f o r m e d r o t o r g i v e s r i s e t o a

s p e c t r u m o f s t a t e s w h o s e e n e r g y e i g e n v a l u e s o b e y a J ( J + 1 ) r u l e . A n u m

b e r o f e x a m p l e s o f c o l l e c t i v e n u c l e a r m o t i o n a r e s k e t c h e d i n f i g u r e 1 . 1 .

B e f o r e 1 9 7 4 , n u c l e a r c o l l e c t i v e m o d e l s w e r e b u i l t u p d i r e c t l y f r o m

g e o m e t r i c d e s c r i p t i o n s o f t h e e q u i l i b r i u m n u c l e a r s h a p e s u c h a s t h a t i n

e q u a t i o n ( 1 . 2 ) . I n t h e m i d - 1 9 7 0 ' s , a n a l t e r n a t i v e a p p r o a c h t o t h e

d e s c r i p t i o n o f n u c l e a r c o l l e c t i v e m o t i o n w a s d e v e l o p e d a r o u n d t h e a l g e

b r a i c p r o p e r t i e s o f t h e d y n a m i c s y m m e t r i e s i n h e r e n t i n t h e q u a d r u p o l e

m o d e s o f t h e n u c l e u s . T h i s m o d e l , t h e I n t e r a c t i n g B o s o n M o d e l ( I B M ) ( A r

7 5 , A r 7 6 , A r 7 8 , A r 7 9 ) , h a s p r o v e n t o b e u s e f u l f o r t h e d e s c r i p t i o n o f

a w i d e r a n g e o f n u c l e a r s t r u c t u r e p h e n o m e n a . S i m i l a r a l g e b r a i c m e t h o d s

h a v e b e e n s u c c e s s f u l l y a p p l i e d t o m o l e c u l a r b e h a v i o r a t b o t h t h e n u c l e a r

a n d a t o m i c l e v e l s ( l a 8 1 a , l a 8 1 b , l a 8 2 a , V a 8 2 ) . T h e g e n e r i c n a m e

a p p l i e d t o e a c h o f t h e s e m o d e l s i s S p e c t r u m G e n e r a t i n g A l g e b r a ( S G A ) .

I n t h e c o n t e x t o f t h e s u c c e s s e s o f t h e s h e l l m o d e l f o r s p h e r i c a l

n u c l e a r s y s t e m s a n d t h e c o l l e c t i v e p i c t u r e f o r n u c l e i r e m o v e d f r o m m a g i c

w h i l e v a l u e s a s l a r g e a s 0 . 1 0 c a n b e f o u n d .

F i g u r e 1 . 1 C l a s s i c a l d e p i c t i o n s o f

c o l l e c t i v e n u c l e a r p h e n o m e n a .

NUCLEAR C O LLE C TIV ITY

FOOTBALL

ROTATIONS

DIPOLE QUADRUPOLE OCTUPOLE

DOORKNOBS NEUTRONS PROTONS NEUTRONS PROTONS

TRANSLATIONS TORSIONS

VIBRATIONS MOLECULAR i.>i

n u m b e r s , t h e d e v e l o p m e n t o f a s h e l l m o d e l f o r d e f o r m e d n u c l e i w a s a

n a t u r a l o n e . N i l s s o n ' s o r i g i n a l d e f o r m e d s h e l l m o d e l c a l c u l a t i o n s ( N i

5 5 ) m e t w i t h g r e a t s u c c e s s i n p r e d i c t i n g t h e s p i n s o f g r o u n d s t a t e s o f

o d d - A n u c l e i a n d c o n t r i b u t e d t o a n u n d e r s t a n d i n g , f r o m a m i c r o s c o p i c

p o i n t o f v i e w , o f m a n y o t h e r o b s e r v a b l e s i n d e f o r m e d n u c l e i .

O v e r t h e l a s t t w e n t y y e a r s , s i g n i f i c a n t d e v e l o p m e n t s i n h e a v y i o n

a c c e l e r a t o r s , h i g h r e s o l u t i o n g a m m a r a y d e t e c t o r s a n d l a r g e s o l i d a n g l e

d e t e c t o r a r r a y s f o r t h e s t u d y o f h i g h m u l t i p l i c i t y g a m m a r a y c a s c a d e s

h a v e s t i m u l a t e d t h e s t u d y o f t h e n u c l e u s a t h i g h a n g u l a r m o m e n t u m ( d e V o

8 3 ) . J u s t a s t h e i n t r o d u c t i o n o f d e f o r m a t i o n i n t h e n u c l e a r s y s t e m

c h a n g e s t h e f o r m o f t h e n u c l e a r s h e l l m o d e l , s o a l s o t h e p l a c e m e n t o f

t h e n u c l e u s i n a r o t a t i n g f r a m e o f r e f e r e n c e m o d i f i e s t h e r e l a t i v e p o s i

t i o n s o f t h e s i n g l e p a r t i c l e o r b i t a l s . T h e m o s t s t r i k i n g e f f e c t o f h i g h

a n g u l a r m o m e n t u m o n t h e n u c l e u s i s t h e p r o g r e s s i v e l o s s o f p a i r i n g c o r

r e l a t i o n s b e t w e e n o p p o s i t e l y r o t a t i n g n u c l e o n s , r e f l e c t i n g C o r i o l i s

e f f e c t s a n d c e n t r i f u g a l l y i n d u c e d d e f o r m a t i o n c h a n g e s . W i t h t h e d e v e l

o p m e n t o f t h e C r a n k e d S h e l l M o d e l b y B e n g t s s o n a n d F r a u e n d o r f i n 1 9 7 9

( B e 7 9 a , B e 7 9 b ) , i t b e c a m e p o s s i b l e t o t r a c e t h e e v o l u t i o n o f t h e p a i r

i n g f o r c e a n d t h e m o m e n t o f i n e r t i a o f t h e n u c l e u s a t h i g h a n g u l a r

m o m e n t a . C o n s e q u e n t l y , a s u b s t a n t i a l k n o w l e d g e o f s i n g l e p a r t i c l e o r b i

t a l s a t h i g h a n g u l a r m o m e n t a w a s g a i n e d .

A l l o f t h i s w o r k , h o w e v e r , r e l i e d o n t h e p r e s e n c e , i n t h e i n t r i n s i c

n u c l e a r s h a p e , o f a s y m m e t r y c o m m o n t o b o t h q u a d r u p o l e a n d h e x a d e c a p o l e

m u l t i p o l a r i t i e s a n d i n d e e d i n h e r e n t i n a l l e v e n m u l t i p o l a r i t i e s . I f a

p l a n e p e r p e n d i c u l a r t o t h e s y m m e t r y a x i s o f t h e q u a d r u p o l e s h a p e i s

d r a w n , i t i s c l e a r t h a t a r e f l e c t i o n t h r o u g h t h i s p l a n e i s a s y m m e t r i c

o n e f o r a n y n u c l e u s w h o s e s h a p e c a n b e d e s c r i b e d e n t i r e l y b y e v e n

s p h e r i c a l h a r m o n i c s . T h e b r e a k i n g o f s u c h a n i n t r i n s i c r e f l e c t i o n s y m

m e t r y i s s i g n a l l e d i n t h e l o w e n e r g y e x c i t a t i o n s p e c t r u m o f a h e a v y

n u c l e u s i n t w o w a y s . T h e f i r s t s i g n a t u r e i s t h e a p p e a r a n c e o f n e g a t i v e

p a r i t y s t a t e s o f a c o l l e c t i v e o r i g i n ( i n s t e a d o f v i a t h e m e c h a n i s m o f

s i n g l e p a r t i c l e p r o m o t i o n w i t h i n t h e s h e l l s t r u c t u r e ) . S u c h s t a t e s c a n

n o t b e g e n e r a t e d b y s h a p e s c o m p o s e d e x c l u s i v e l y o f e v e n m u l t i p o l a r i t i e s .

T h e s e c o n d s i g n a t u r e f o r r e f l e c t i o n a s y m m e t r y i n h e a v y n u c l e i i s t h e

p r e s e n c e o f u n u s u a l l y s t r o n g g a m m a r a y t r a n s i t i o n s o f o d d e l e c t r i c m u l

t i p o l a r i t i e s , m o s t i m p o r t a n t l y d i p o l e , b e t w e e n q u a n t u m s t a t e s o f t h e

n u c l e u s . T h e t w o m e c h a n i s m s f o r p r o d u c i n g r e f l e c t i o n a s y m m e t r i c s h a p e s

w h i c h w i l l b e c o n s i d e r e d i n t h i s w o r k l e a d t o t h e s e e n h a n c e d e l e c t r i c

d i p o l e ( E l ) t r a n s i t i o n s i n d i f f e r e n t w a y s .

S t r o n g E l t r a n s i t i o n s a r e p a r t i c u l a r l y s i g n i f i c a n t g i v e n t h e

a b s e n c e o f t h e d i p o l e t e r m i n t h e e x p r e s s i o n f o r t h e n u c l e a r s u r f a c e

( 1 . 2 ) . B e c a u s e o f t h i s a b s e n c e , t h e e l e c t r i c d i p o l e o p e r a t o r i s d e p e n

d e n t i n f i r s t o r d e r u p o n t h e s e p a r a t i o n o f t h e c e n t e r o f n u c l e a r c h a r g e

f r o m t h e c e n t e r o f m a s s . S u c h a s e p a r a t i o n o c c u r s i n t h e g i a n t d i p o l e

r e s o n a n c e o f t h e n u c l e u s , a p h e n o m e n o n w h i c h i s f o u n d a t e x c i t a t i o n

e n e r g i e s o f 1 5 M e V a n d a b o v e ( G o 4 8 ) ; i n t h i s m o d e , t h e p r o t o n a n d n e u

t r o n f l u i d s o s c i l l a t e b a c k a n d f o r t h c o h e r e n t l y a n d c o l l e c t i v e l y w i t h

r e s p e c t t o o n e a n o t h e r ( s e e f i g u r e 1 . 1 ) . T h e c e n t e r o f m a s s r e m a i n s

s t a t i o n a r y a t t h e g e o m e t r i c c e n t e r o f t h e n u c l e u s , b u t t h e c e n t e r o f

c h a r g e i s f o u n d a t t h e c e n t e r o f t h e p r o t o n f l u i d , w h i c h o s c i l l a t e s r e l

a t i v e t o t h e c e n t e r o f m a s s .

-6-

In parallel, the study of molecular resonances in heavy ion inter-

a c t i o n s l e d t o t h e d i s c o v e r y t h a t i n t h e c a s e s o f 1 2 C + 1 2 C a n d 1 2 C + 1 6 0

r e s o n a n c e s a t e n e r g i e s c l o s e t o t h e C o u l o m b b a r r i e r ( t h e e n e r g y r e q u i r e d

t o o v e r c o m e t h e e l e c t r o s t a t i c r e p u l s i o n b e t w e e n t w o n u c l e i ) , t h e d a t a

c o u l d b e c o r r e l a t e d i n t h e f r a m e w o r k o f a n S G A i n w h i c h t h e v e c t o r c o n

n e c t i n g t h e c e n t e r s o f t h e t w o i o n s i s t r e a t e d a l g e b r a i c a l l y ( E r 8 1 , l a

8 1 a , R u 8 4 ) . I n t h i s t r e a t m e n t , c a l l e d t h e V i b r o n m o d e l , t h e s e p a r a t i o n

v e c t o r c o n s t i t u t e s a d i p o l e d e g r e e o f f r e e d o m . T h i s m o d e l a l s o m a k e s

t h e i n t u i t i v e l y p l a u s i b l e p r e d i c t i o n o f e n h a n c e d E l t r a n s i t i o n s b e t w e e n

m o l e c u l a r s t a t e s o f o p p o s i t e p a r i t y i n a n o n - s e l f c o n j u g a t e d i n u c l e a r

s y s t e m , a s y s t e m i n w h i c h t h e c h a r g e - t o - m a s s r a t i o s o f t h e t w o c o m p o

n e n t s o f t h e m o l e c u l e a r e d i f f e r e n t ( c o n s e q u e n t l y s e p a r a t i n g t h e c e n t e r

o f c h a r g e o f t h e s y s t e m f r o m t h e c e n t e r o f m a s s a n d g i v i n g t h e m o l e c u l e

a n e l e c t r i c d i p o l e m o m e n t ) . H o w e v e r , n e g a t i v e p a r i t y m o l e c u l a r s t a t e s

d o n o t e x i s t i n 1 2 C + 1 2 C , a n d t h e s e l f - c o n j u g a t e n a t u r e o f t h e 1 2 C + 1 2 0

s y s t e m d i s a l l o w s E l t r a n s i t i o n s t o ' f i r s t o r d e r . T h i s l e d G a i a n d c o l

l a b o r a t o r s t o e x a m i n e l i g h t n u c l e a r s y s t e m s w i t h n o n - z e r o i s o s p i n i n a

s e a r c h f o r t h e p r e d i c t e d e n h a n c e d E l t r a n s i t i o n s . G a i e t a l . o b s e r v e d

t h a t E l t r a n s i t i o n s i n 1 8 0 a n d 1 0 B e w e r e b y f a r t h e s t r o n g e s t i n e v e n

m a s s l i g h t n u c l e i , a n d s u b s e q u e n t l y m a d e a d e t a i l e d s p e c t r o s c o p i c s t u d y

o f 1 8 0 v i a 1 4 C ( a , 2 T ) 1 8 0 a n d 1 4 C ( 7 L i , t j f ) 1 8 0 r e a c t i o n s w i t h t h e r e s u l t s

s h o w n i n f i g u r e 1 . 2 ( G a 8 3 a , R u 8 4 ) .

T h e i m p o r t a n c e o f p r e - f o r m e d a l p h a p a r t i c l e s i n t h e c o l l e c t i v e

b e h a v i o r o f l i g h t n u c l e i ( A < 3 0 ) h a s b e e n k n o w n s i n c e t h e 1 9 5 0 ' s ; t h i s

c l u s t e r i n g b e h a v i o r c a n l e a d t o b o u n d n o n - s e l f c o n j u g a t e d i n u c l e a r

m o l e c u l a r s y s t e m s . A s a n e x a m p l e w e c a n c o n s i d e r t h e n u c l e u s 1 8 0 , i n

w h i c h t h e d i n u c l e a r a l p h a p a r t i c l e c l u s t e r i n g c o n f i g u r a t i o n w o u l d b e

F i g u r e 1 . 2 A p r o p o s e d a l p h a p a r t i

c l e c l u s t e r m o l e c u l a r b a n d i n 1 8 0 ( G a

8 3 a ) .

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4H e + 14C . F i g u r e 1 . 2 s h o w s a p r o p o s e d b a n d o f s t a t e s i n 180 i n w h i c h

t h i s d i n u c l e a r a l p h a p a r t i c l e c l u s t e r c o n f i g u r a t i o n m ay p l a y a m a j o r

r o l e ( G a 8 3 a ) . T h i s b a n d d e m o n s t r a t e s t h e tw o s i g n a t u r e s o f r e f l e c t i o n

a s y m m e t r y t h a t we h a v e d i s c u s s e d , l o w l y i n g n e g a t i v e p a r i t y s t a t e s a n d

e n h a n c e d i n t r a b a n d E l t r a n s i t i o n s . A t h i r d s i g n a t u r e t h a t f o l l o w s

d i r e c t l y f r o m t h e a l p h a p a r t i c l e c l u s t e r i n g c o n c e p t i s t h e p r e s e n c e o f

l a r g e a l p h a p a r t i c l e d e c a y w i d t h s . T h i s s i g n a t u r e i s q u i t e s t r i k i n g l y

d e m o n s t r a t e d i n 180 , w h e r e t h e s e w i d t h s h a v e b e e n m e a s u r e d v i a r e s o n a n t

a l p h a p a r t i c l e s c a t t e r i n g ( A j 8 3 ) .

I t b e a r s e m p h a s i s , h o w e v e r , t h a t i n n u c l e i s u c h a s 180 , t h i s c l u s

t e r s t r u c t u r e c o e x i s t s w i t h b o t h s h e l l a n d m o re common r e f l e c t i o n sym m e

t r i c c o l l e c t i v e s t r u c t u r e a n d t h a t r e s i d u a l i n t e r a c t i o n s r e s u l t i n

a d m i x t u r e s o f a l l t h r e e i n d i f f e r e n t q u a n t u m s t a t e s . T h i s h a s t h e c o n

s e q u e n c e t h a t a l t h o u g h t h e s i g n a t u r e s c h a r a c t e r i s t i c o f e a c h o f t h e

a b o v e m e n t i o n e d s t r u c t u r e s c a n b e i d e n t i f i e d , t h e y w o u l d n o t b e e x p e c t e d

t o a p p e a r i n t h e c l a s s i c a l f o r m t h a t w o u l d b e p r e s e n t w e r e o n l y a s i n g l e

t y p e o f s t r u c t u r e i n v o l v e d .

P r o m p t e d b y t h e o b s e r v a t i o n o f n u c l e a r d e c a y i n r a r e e a r t h a n d

a c t i n i d e n u c l e i t h r o u g h t h e e m i s s i o n o f a n a l p h a p a r t i c l e , a d e b a t e h a s

c o n t i n u e d t o t h i s d a y a b o u t w h e t h e r a l a r g e p r o b a b i l i t y f o r a l p h a p a r t i

c l e d e c a y n e c e s s a r i l y i m p l i e s t h e e x i s t e n c e o f p r e - f o r m e d a l p h a p a r t i

c l e s a t t h e n u c l e a r s u r f a c e . F i g u r e 1 . 3 d i s p l a y s r e d u c e d g r o u n d s t a t e

a l p h a p a r t i c l e d e c a y w i d t h s ( d e d u c e d f r o m m e a s u r e d l i f e t i m e s ) f o r h e a v y

n u c l e i a c r o s s t h e p e r i o d i c t a b l e . T h e l a r g e s t a l p h a p a r t i c l e d e c a y

w i d t h s a r e f o u n d j u s t a b o v e t h e n e u t r o n m a g i c n u m b e r s 82 a n d 1 2 6 .

T h e r e f o r e , i t w o u l d b e i n t u i t i v e l y p r o b a b l e t o f i n d a l p h a p a r t i c l e c l u s -

F i g u r e 1 . 3 C o m p i l a t i o n o f r e d u c e d

a l p h a p a r t i c l e d e c a y w i d t h s f r o m ( R o

8 3 ) . T h e 2 1 8 R a w i d t h h a s b e e n

a d j u s t e d t o c o n f o r m t o t h e r e c e n t

r e s u l t o f ( R a 8 6 ) .

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I t i s i m m e d i a t e l y a p p a r e n t t h a t s u c h a c o n f i g u r a t i o n n e c e s s a r i l y

b r e a k s r e f l e c t i o n s y m m e t r y ( s e e f i g u r e 1 . 4 ) . F u r t h e r , t h i s c o n f i g u r a

t i o n , i n a h e a v y n u c l e u s , w o u l d g i v e a n i n t r i n s i c e l e c t r i c d i p o l e m om ent

i n a p a r t i c u l a r l y s i m p l e w a y . T h e a l p h a p a r t i c l e h a s tw o p r o t o n s a n d

tw o n e u t r o n s , g i v i n g a c h a r g e t o m a s s r a t i o o f 1 / 2 . T h e h e a v y c o r e

n u c l e u s , h o w e v e r , a l w a y s h a s m o re n e u t r o n s t h a n p r o t o n s a n d a c h a r g e t o

m a s s r a t i o o f l e s s t h a n 1 / 2 . T h e r e f o r e , t h e c e n t e r o f c h a r g e o f t h e

n u c l e u s w i l l b e c l o s e r t o t h e a l p h a p a r t i c l e c l u s t e r t h a n t h e c e n t e r o f

m a s s . T h e h i g h e r e l e c t r i c m u l t i p o l a r i t i e s o f s u c h a c o n f i g u r a t i o n c a n

b e s i m p l y s h o w n t o b e n o n - z e r o b y a c l a s s i c a l c a l c u l a t i o n ( A l 8 2 ) .

T h e a d d i t i o n o f a n o c t u p o l e c o m p o n e n t t o t h e n u c l e a r s h a p e a l s o

r e s u l t s i n t h e b r e a k i n g o f r e f l e c t i o n s y m m e t r y ( s e e f i g u r e 1 . 4 ) a n d

g i v e s r i s e t o s t r o n g e l e c t r i c o c t u p o l e ( E 3 ) t r a n s i t i o n s i n a s t r a i g h t

f o r w a r d m a n n e r . B e c a u s e t h e f i r s t o r d e r e l e c t r o m a g n e t i c t r a n s i t i o n

o p e r a t o r s i m p l y c o n s i s t s o f t h e c l a s s i c a l e x p r e s s i o n f o r t h e e l e c t r o m a g

n e t i c m o m e n t , a n e l e c t r i c o c t u p o l e m om ent i s r e f l e c t e d i n t h e o b s e r v a

t i o n o f s t r o n g e l e c t r i c o c t u p o l e t r a n s i t i o n s b e t w e e n q u a n t u m s t a t e s .

I n t h e l a t e 1 9 5 0 ' s , tw o s e p a r a t e t r e a t m e n t s o f t h e o c t u p o l e s h a p e ,

o n e b y B o h r a n d M o t t e l s o n ( B o 5 7 , B o 5 8 ) , t h e o t h e r b y S t r u t i n s k y ( S t

5 7 ) , s h o w e d t h a t t h i s s h a p e c o m p o n e n t w o u l d l e a d t o a l a r g e r d e n s i t y o f

p r o t o n s r e l a t i v e t o n e u t r o n s i n o n e e n d o f t h e n u c l e u s t h a n i n t h e

o t h e r . C o n s e q u e n t l y , t h e c e n t e r o f c h a r g e w o u l d b e d i s p l a c e d f r o m t h e

c e n t e r o f m a s s a n d a n e l e c t r i c d i p o l e m o m e n t , a s w e l l a s s t r o n g E l t r a n

s i t i o n s , w o u l d r e s u l t . T h e c a l c u l a t i o n s o f r e f e r e n c e s ( B o 5 7 , B o 5 8 ) a n d

( S t 5 7 ) a r e b a s e d , h o w e v e r , o n d i f f e r e n t e l e c t r o s t a t i c e f f e c t s . I n t h e

-11-tering configurations in these regions.

F i g u r e 1 . 4 A l p h a c l u s t e r ( a ) a n d

o c t u p o l e d e f o r m e d ( b ) c o n f i g u r a t i o n s

f o r 2 2 0 R a , d r a w n t o s c a l e . T h e o c t u p o l e d e f o r m e d s h a p e h a s t h e d e f o r m a

t i o n p a r a m e t e r s p2 = 0 . 1 2 a n d 33 = 0 . 0 8

p r e d i c t e d i n ( N a 8 4 b ) f o r 2 2 2 R a .

B o h r a n d M o t t e l s o n t r e a t m e n t , i t i s a s s u m e d t h a t a u n i f o r m

p r o t o n - n e u t r o n d i s t r i b u t i o n w o u l d e x i s t i n a s p h e r i c a l n u c l e u s , a n d t h e

p o l a r i z a t i o n a r i s e s e x c l u s i v e l y f r o m t h e r e f l e c t i o n a s y m m e t r i c d e f o r m a

t i o n . I n c o n t r a s t , S t r u t i n s k y t a k e s i n t o a c c o u n t t h e m o n o p o l e e l e c t r o s

t a t i c f i e l d , w h i c h w o u l d f o r c e t h e p r o t o n s t o w a r d t h e s u r f a c e i n a

s p h e r i c a l n u c l e u s . C o n s e q u e n t l y , t h e tw o c a l c u l a t i o n s r e s u l t i n p r e

d i c t i o n s o f t h e e l e c t r i c d i p o l e m om ent w h i c h h a v e o p p o s i t e s i g n s , - t h e

c u r r e n t w e i g h t o f t h e o r e t i c a l e v i d e n c e s u g g e s t s t h a t t h e S t r u t i n s k y

r e s u l t i s t h e c o r r e c t o n e . A v e r y r e c e n t c a l c u l a t i o n ( D o 8 6 ) a d d s t h e

e f f e c t s o f a p o s t u l a t e d " n e u t r o n s k i n " o n t h e n u c l e u s . T h e r e l a t i v e

i m p o r t a n c e o f t h e s e t h r e e t e r m s h a s n o t b e e n d e t e r m i n e d , a n d p r e s e n t s a

f o r m i d a b l e t h e o r e t i c a l o b s t a c l e .

J u s t a s f o r s h a p e c o m p o n e n t s o f o t h e r m u l t i p o l a r i t i e s , t h e o c t u p o l e

m o m e n t c a n b e m a n i f e s t e d i n b o t h v i b r a t i o n a l a n d s t a t i c f o r m s . O c t u p o l e

v i b r a t i o n a l b e h a v i o r h a s b e e n o b s e r v e d t h r o u g h o u t t h e p e r i o d i c t a b l e ,

a n d i s a s s o c i a t e d w i t h e n h a n c e d E l t r a n s i t i o n s i n t h e r e g i o n n e a r Z = 6 4 .

T h e r e i s s t r o n g e v i d e n c e ( s e e s e c t i o n 2 . 2 ) t h a t t h e a l t e r n a t i n g p a r i t y

s t r u c t u r e i n t h e n u c l e u s 1 50G d ( f i g u r e 1 . 5 ) r e s u l t s f r o m t h e c o u p l i n g o f

a n o c t u p o l e p h o n o n c a r r y i n g s p i n - p a r i t y o f 3 “ t o t h e p o s i t i v e p a r i t y

s t a t e s o f t h e g r o u n d s t a t e b a n d o f t h i s q u a d r u p o l e v i b r a t i o n a l n u c l e u s

( H a 7 7 ) . I f a n o c t u p o l e p h o n o n i s c o u p l e d t o a q u a d r u p o l e d e f o r m e d

n u c l e u s , a n a d d i t i o n a l 1 “ s t a t e c o u l d b e l o c a t e d b e l o w t h e 3 “ s t a t e .

G a m m a - r a y b r a n c h i n g r a t i o s s u g g e s t t h a t E l t r a n s i t i o n s f r o m t h e r e s u l t

i n g n e g a t i v e p a r i t y s t a t e s a r e s i g n i f i c a n t l y s t r o n g e r t h a n t h o s e a r i s i n g

f r o m t h e t r a n s i t i o n o f a s i n g l e p a r t i c l e f r o m o n e s h e l l m o d e l o r b i t a l t o

a n o t h e r .

-13-

F i g u r e 1 . 5 A p a r t i a l l e v e l s p e c

t r u m o f 15DG d (H a 7 7 ) . T h e n e g a t i v e

p a r i t y s t a t e s s h o w n h a v e b e e n i n t e r

p r e t e d i n ( H a 7 7 ) t o b e t h e r e s u l t o f

t h e c o u p l i n g o f a n o c t u p o l e p h o n o n

h a v i n g J 1 " t o t h e q u a d r u p o l e v i b r a

t i o n a l c o r e .

-14-

We w o u l d a l s o e x p e c t t o o b s e r v e l o w - l y i n g n e g a t i v e p a r i t y s t a t e s

a n d e n h a n c e d E l t r a n s i t i o n s i n a s t a t i c a l l y o c t u p o l e d e f o r m e d n u c l e u s .

T h e n e g a t i v e p a r i t y b a n d w o u l d h a v e a 1 " s t a t e a s t h e b a n d h e a d , a n d

w o u l d f o r m a n a l t e r n a t i n g p a r i t y s e q u e n c e a t h i g h e r s p i n s . N o o b s e r v a

t i o n s o f s t a t i c o c u t p o l e d e f o r m a t i o n i n t h e g r o u n d s t a t e o f a n u c l e u s

h a v e y e t b e e n c o n f i r m e d .

T h e s t u d y o f r e f l e c t i o n a s y m m e t r i c s h a p e s a l s o g i v e s u s t h e o p p o r

t u n i t y t o e x a m in e o u r i d e a s c o n c e r n i n g s i n g l e p a r t i c l e m o t i o n a n d c o l

l e c t i v e b e h a v i o r i n s y s t e m s w i t h d i f f e r e n t s y m m e t r i e s f r o m t h o s e w i t h

w h i c h m o d e l s s u c h a s t h e B o h r - M o t t e l s o n c o l l e c t i v e m o d e l , t h e I n t e r a c t

i n g B o s o n M o d e l , t h e N i l s s o n d e f o r m e d s h e l l m o d e l a n d t h e C r a n k e d S h e l l

M o d e l w e r e o r i g i n a l l y d e v e l o p e d . I n t h e p a s t s e v e r a l y e a r s a c o n s i d e r a

b l e t h e o r e t i c a l e f f o r t h a s b e e n m o u n t e d i n o r d e r t o g e n e r a l i z e t h e s e

c o n c e p t s t o t h e c a s e i n w h i c h r e f l e c t i o n s y m m e t r y i s b r o k e n ( N a 8 5 , Ro

7 8 , R o 8 2 a , E n 8 5 , D a 8 3 , l a 8 2 b ) . T h e s e e f f o r t s h a v e b e e n s y s t e m a t i

c a l l y h a n d i c a p p e d b y t h e l a c k o f t h e l a r g e b o d y o f s y s t e m a t i c d a t a o n

r e f l e c t i o n a s y m m e t r i c s y s t e m s n e e d e d t o r e f i n e t h e s e i d e a s i n t o r e l i a b l e

m o d e l s .

F o r o v e r t h i r t y y e a r s i t h a s b e e n k n o w n t h a t r e f l e c t i o n a s y m m e t r i c

s h a p e s m u s t p l a y a n i m p o r t a n t r o l e a t v e r y l o w e x c i t a t i o n e n e r g i e s i n a

n u m b e r o f i s o t o p e s o f R a a n d T h . T h e f i r s t e v i d e n c e t h a t t h i s m i g h t be

t h e c a s e w a s f o u n d b y S t e p h e n s a n d c o l l a b o r a t o r s a t B e r k e l e y i n t h e

m i d - 1 9 5 0 ' s ( S t 5 4 , S t 5 5 , S t 5 7 ) . I n s p e c t r o s c o p i c s t u d i e s o f t h e r a d i

o a c t i v e d e c a y c h a i n s o f s e v e r a l U i s o t o p e s , i t w a s d i s c o v e r e d t h a t t h e

a l p h a p a r t i c l e d e c a y s o f e v e n - m a s s U a n d T h i s o t o p e s p o p u l a t e d n e g a t i v e

p a r i t y s t a t e s o f v e r y l o w e n e r g i e s i n t h e d a u g h t e r n u c l e i . T h e s e d e c a y

-15-

s t u d i e s w e r e e x p a n d e d a t B e r k e l e y u n t i l t h e e a r l y 1 9 6 0 ' s ( R u 6 1 ) , a n d

f u r t h e r r e f i n e d i n t h e s t u d i e s o f K u r c e w i c z ( K u 7 6 , Ku 7 7 , K u 7 8 ) .

F i g u r e 1 . 6 d i s p l a y s r e s u l t s p r e s e n t e d b y K u r c e w i c z o n t h e d e c a y

c h a i n o f 2 3 0 U ( K u 7 6 ) . T h e l o w - l y i n g n e g a t i v e p a r i t y s t a t e s a r e f o u n d

i n b o t h 2 2 6 T h a n d 2 2 2 R a a t r o u g h l y e q u a l e n e r g i e s . T e n t a t i v e a s s i g n

m e n t s h a v e b e e n g i v e n t o t h r e e s t a t e s s t a t e s i n 2 1 8 R n a t 6 5 3 k e V , 797

k e V a n d 8 4 0 k e V ; t h e s e r e q u i r e som e e x p l a n a t i o n . I n t h e o r i g i n a l w o r k

( K u 7 6 ) , n o s u c h a s s i g n m e n t s w e r e r e p o r t e d . I t w a s o b s e r v e d i n ( P e 8 1 ) ,

h o w e v e r , t h a t t h e gamma r a y d e e x c i t a t i o n s o f t h e s e s t a t e s s t r o n g l y s u g

g e s t e d t h e a s s i g n m e n t s s h o w n i n f i g u r e 1 . 6 . F u r t h e r , e l e c t r o n c o n v e r

s i o n d a t a ( L e 6 3 ) c o n f i r m s t h e p a r i t y a s s i g n m e n t m ade f o r t h e 7 9 7 k e V

l e v e l . I f t h e s e a s s i g n m e n t s a r e c o r r e c t , t h e n e g a t i v e p a r i t y s t a t e s i n

2 1 8 R n a r e s i g n i f i c a n t l y h i g h e r i n e n e r g y t h a n t h o s e i n 2 2 2 R a a n d 2 2 6 T h .

A n o t h e r c h a r a c t e r i s t i c o f t h e s e R n s t a t e s t h a t d i s t i n g u i s h e s th e m f r o m

c o r r e s p o n d i n g s t a t e s i n t h e p a r e n t a n d g r a n d p a r e n t n u c l e i i s t h e i r

o r d e r i n g . I n b o t h 2 2 2 R a a n d 2 2 6 T h t h e 1 " s t a t e i s l o c a t e d a b o u t 80 k e V

b e l o w t h e 3 " s t a t e . H o w e v e r , t h e 3 ' s t a t e f a l l s 43 k e V b e l o w t h e 1 "

s t a t e i n 2 1 8 R n . S u c h i n f o r m a t i o n m ay b e h e l p f u l i n i n v e s t i g a t i o n s o f

d e t a i l s o f t h e n u c l e a r s h a p e .

R e c e n t l y , s e v e r a l i n v e s t i g a t o r s h a v e s t u d i e d t h e h i g h s p i n s t r u c

t u r e a s s o c i a t e d w i t h t h e s e l o w - l y i n g n e g a t i v e p a r i t y s t a t e s . A n e x a m p le

o f t h i s h i g h s p i n s t r u c t u r e i s t h a t o f 2 1 8 R a s h o w n i n f i g u r e 1 . 7 . Two

f e a t u r e s s t a n d o u t i n t h i s n u c l e u s . T h e f i r s t i s t h e a l t e r n a t i n g p a r i t y

b a n d s t r u c t u r e a t s p i n s o f 4 R a n d a b o v e . T h e s e c o n d f e a t u r e , n o t s h o w n

i n t h e f i g u r e , i s t h e v e r y s t r o n g e n h a n c e m e n t o f t h e E l t r a n s i t i o n s

l i n k i n g t h e o p p o s i t e p a r i t y m e m b e r s o f t h e g r o u n d s t a t e b a n d . I t h a s

-16-

F i g u r e 1 . 6 A l p h a p a r t i c l e d e c a y

c h a i n o f 2 3 0 U f r o m ( K u 7 6 ) . T e n t a

t i v e s p i n a s s i g n m e n t s i n 2 1 8 R n a r e

f r o m ( P e 8 1 ) .

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b e e n d e t e r m i n e d t h a t t h e r e d u c e d m a t r i x e l e m e n t s o f t h e s e t r a n s i t i o n s

a r e t h e l a r g e s t ( a f t e r m a s s d e p e n d e n c e h a s b e e n r e m o v e d ) f o u n d i n h e a v y

e v e n - m a s s n u c l e i ( G a 8 3 b ) .

T h e f i r s t o b j e c t i v e o f t h e p r e s e n t w o r k i s a n e x a m i n a t i o n o f t h e

q u e s t i o n o f r e f l e c t i o n a s y m m e t r y i n e v e n R n , R a a n d T h n u c l e i t h r o u g h

t h e s t u d y o f t h e s y s t e m a t i c b e h a v i o r o f t h e i r E l m a t r i x e l e m e n t s a s w e l l

a s t h e i r e x c i t a t i o n s p e c t r a a t b o t h l o w a n d h i g h s p i n s . T h e f i r s t w a y

i n w h i c h we w i l l d o t h i s i s v i a a n e x p e r i m e n t a l s t u d y o f t h e gamma r a y

s p e c t r a o f 2 1 6 R n a n d 2 2 0 R a p r o d u c e d i n t h e 14C + 2 0 8 P b f u s i o n - e v a p o r a t i o n

r e a c t i o n a t beam e n e r g i e s o f 60 - 70 M e V . E a c h o f t h e s e n u c l e i i s tw o

n u c l e o n s a w a y f r o m 2 1 8 R a , a n d i n f o r m a t i o n g a i n e d f r o m th e m w i l l c l a r i f y

t h e s y s t e m a t i c b e h a v i o r o f o b s e r v a b l e s i n t h e m a s s r e g i o n v e r y c l o s e t o

2 1 8 R a .

T h e s e c o n d s t u d y w i l l i n v o l v e t h e t e s t i n g o f n u c l e a r c o l l e c t i v e

m o d e l s c o n s t r u c t e d a r o u n d t h e s h a p e s d e s c r i b e d a b o v e f r o m d a t a a v a i l a b l e

o n R a a n d T h i s o t o p e s . T h i s w i l l i n c l u d e a d i s c u s s i o n o f p r e d i c t i o n s o f

a N i l s s o n M o d e l a n d a C r a n k e d S h e l l M o d e l (N a 8 5 ) , e a c h o f w h i c h

i n c l u d e s a n o c t u p o l e d e g r e e o f f r e e d o m (N a 8 5 ) , a s w e l l a s e x a m i n a t i o n

o f c a l c u l a t i o n s o f t h e p o t e n t i a l e n e r g y o f t h e n u c l e a r g r o u n d s t a t e f o r

a r a n g e o f s h a p e s ( L e 8 2 a , N a 8 4 b ) . We w i l l a l s o p r e s e n t t h e r e s u l t s o f

a c a l c u l a t i o n o f e n e r g y l e v e l s p e c t r a f o r f o u r R a i s o t o p e s i n w h i c h t h e

V i b r o n f o r m a l i s m ( D a 8 3 ) i s a p p l i e d t o t h e a l p h a p a r t i c l e c l u s t e r i n g

p i c t u r e .

I n a b r o a d e r w a y , we w i l l a l s o e x a m in e t h e r e l a t i o n s h i p o f r e f l e c

t i o n a s y m m e t r i c b e h a v i o r i n R n , R a a n d T h n u c l e i t o t h e b e t t e r u n d e r

s t o o d o c t u p o l e v i b r a t i o n a l b e h a v i o r i n P b , P o , U a n d P u i s o t o p e s , a s

-19-

w e l l a s i n a r a n g e o f e l e m e n t s i n t h e 5 6 < Z < 8 2 , 8 2 < N < 1 2 6 r e g i o n . B y

c o m p a r i n g t h e s y s t e m a t i c b e h a v i o r o f e n e r g y e x c i t a t i o n s p e c t r a i n t h e s e

r e g i o n s , we h o p e t o e v o l v e new i n s i g h t s i n t o t h e n a t u r e o f t h e r e f l e c

t i o n a s y m m e t r y i n R n , R a a n d T h .

T h e s e c o n d o b j e c t i v e h e r e i s t o s t u d y t h e s t r e n g t h o f t h e c o u p l i n g

b e t w e e n a n u n p a i r e d v a l e n c e n u c l e o n a n d t h e e v e n - e v e n c o r e i n o d d - A n u c

l e i o f t h i s r e g i o n . T h e p a r t i c l e - c o r e c o u p l i n g i s q u i t e d e p e n d e n t o n

t h e d e g r e e o f d e f o r m a t i o n i n t h e c o r e . I n t h e c a s e o f a n i d e a l s p h e r i

c a l n u c l e u s , t h e u n p a i r e d n u c l e o n a n d t h e c o r e i n t e r a c t i n s u c h a w a y

t h a t t h e s p e c t r u m o f t h e o d d - A n u c l e u s c o n s i s t s s i m p l y o f a s e q u e n c e o f

d e g e n e r a t e m u l t i p l e t s , o n e m u l t i p l e t f o u n d a t t h e e n e r g y o f e a c h s t a t e

o f t h e e v e n c o r e n u c l e u s . T h e s t a t e s o f e a c h m u l t i p l e t a r e g e n e r a t e d

t h r o u g h t h e a d d i t i o n o f t h e a n g u l a r m om entum o f t h e u n p a i r e d n u c l e o n ,

c o m m o n ly d e n o t e d b y j , t o t h a t o f t h e c o r r e s p o n d i n g c o r e s t a t e :

ii>(J , j , J ) — | J > x | j > r v c o r e J c o r e J

f o r |J - j| < J < J + j . c o r e c o r e

T h i s s i t u a t i o n i s k n o w n a s t h e w e a k c o u p l i n g l i m i t . I f t h e n u c l e u s h a s

a s m a l l d e f o r m a t i o n , t h e d e g e n e r a c y i s b r o k e n a n d t h e s t a t e s a r e s p r e a d

o v e r a s m a l l e n e r g y r a n g e , g e n e r a l l y a s m a l l f r a c t i o n o f t h e e n e r g y o f

t h e c o r e s t a t e . A s t h e d e f o r m a t i o n b e c o m e s p r o g r e s s i v e l y l a r g e r , p r o

f o u n d c h a n g e s o c c u r . S t a t e s f r o m d i f f e r e n t m u l t i p l e t s a r e m i x e d

t o g e t h e r a n d t h e w e a k c o u p l i n g f o r m a l i s m b e c o m e s i n a p p r o p r i a t e . F o r

l a r g e d e f o r m a t i o n s , t h e s t r o n g c o u p l i n g a p p r o a c h u s e d i n t h e N i l s s o n

d e f o r m e d s h e l l m o d e l i s c o r r e c t . I n t h i s a p p r o a c h , d e f o r m e d c o r e s t a t e s

a r e c o u p l e d t o s i n g l e p a r t i c l e e i g e n s t a t e s o f a d e f o r m e d p o t e n t i a l a n d

-20-

t h e s i n g l e p a r t i c l e e n e r g y e i g e n v a l u e s d e p e n d d i r e c t l y o n t h e c o r e

d e f o r m a t i o n . F u r t h e r , t h e p a r t i c l e - c o r e i n t e r a c t i o n h a m i l t o n i a n h a s a

l a r g e m a g n i t u d e .

I n t h e p a r t i c l e - c o r e s e n s e , 2 1 9 R a c a n b e r e g a r d e d a s a n u n p a i r e d

n e u t r o n c o u p l e d t o a 2 1 8 R a c o r e . B e c a u s e 2 1 8 R a b e h a v e s a s a n e a r l y

s p h e r i c a l n u c l e u s w i t h v i b r a t i o n a l e x c i t a t i o n m o d e s , we w o u l d e x p e c t

f r o m o u r k n o w l e d g e o f v i b r a t i o n a l n u c l e i i n o t h e r r e g i o n s o f t h e p e r i

o d i c t a b l e t h a t 2 1 9 R a w o u l d d i s p l a y p r o p e r t i e s r a t h e r c l o s e t o t h o s e

s e e n i n t h e w e a k c o u p l i n g l i m i t . We t e s t t h a t e x p e c t a t i o n b y e x a m i n i n g

t h e e n e r g y s p e c t r u m o f 2 1 8 R a b y gamma r a y s p e c t r o s c o p y o f s t a t e s p o p u

l a t e d i n f u s i o n - e v a p o r a t i o n r e a c t i o n s o f t h e 14C + 2 0 8 P b s y s t e m . We a l s o

r e v i e w t h e s y s t e m a t i c b e h a v i o r o f o d d - A n u c l e i i n t h e i m m e d i a t e v i c i n i t y

o f 2 1 9 R a i n o r d e r t o e x a m in e t h e p a r t i c l e - c o r e c o u p l i n g s t r e n g t h i n

n u c l e i o f t h i s r e g i o n .

C h a p t e r tw o o f t h i s t h e s i s g i v e s a s c h e m a t i c o v e r v i e w o f t h e o r e t i

c a l a p p r o a c h e s t a k e n i n t h e t r e a t m e n t o f n u c l e i o f t h e R n - R a - T h r e g i o n .

T e c h n i q u e s u s e d i n t h e e x p e r i m e n t a l s t u d i e s p r e s e n t e d i n t h i s w o r k a r e

d e s c r i b e d i n c h a p t e r t h r e e , w h i c h i s f o l l o w e d i n c h a p t e r f o u r b y a p r e s

e n t a t i o n o f t h e e x p e r i m e n t a l r e s u l t s . S t u d i e s o f s y s t e m a t i c b e h a v i o r

a n d d i s c u s s i o n s o f o u r e x p e r i m e n t a l r e s u l t s w i t h i n t h e f r a m e w o r k s o f

s e v e r a l t h e o r e t i c a l p i c t u r e s a r e t o b e f o u n d i n c h a p t e r f i v e . O u r

e x p e r i m e n t a l f i n d i n g s a n d c o n c l u s i o n s a r e s u m m a r i z e d i n c h a p t e r s i x .

B r i e f l y , t h r o u g h o u r s t u d i e s o f 2 1 6 R n a n d 2 2 0 R a we h a v e f o u n d t h a t

t r e n d s i n E l m a t r i x e l e m e n t s a n d a l p h a p a r t i c l e d e c a y w i d t h s a r e c o r r e

l a t e d , s u g g e s t i n g t h a t t h e e n h a n c e d E l t r a n s i t i o n s a n d l a r g e a l p h a p a r

t i c l e d e c a y w i d t h s o b s e r v e d i n t h i s r e g i o n r e s u l t f r o m a s i n g l e n u c l e a r

-21-

p h e n o m e n o n . F u r t h e r , 2 1 8 R a i s t h e l i g h t e s t m em ber o f b o t h i s o t o p i c a n d

i s o t o n i c c h a i n s w h i c h d i s p l a y s c o l l e c t i v e n e g a t i v e p a r i t y s t a t e s a l o n g

t h e y r a s t l i n e . F i n a l l y , 2 1 9 R a a n d l i g h t e r o d d - A n e i g h b o r s c a n b e w e l l

d e s c r i b e d w i t h i n a n a p p r o a c h b a s e d o n w e a k c o u p l i n g ; h o w e v e r , i t i s

l i k e l y t h a t t h i s i s n o t t h e c a s e f o r o d d - A n u c l e i o f m a s s g r e a t e r t h a n

219.

We h a v e f o u n d c l e a r e v i d e n c e f o r new r e f l e c t i o n a s y m m e t r i c s h a p e s

i n t h e l i g h t a c t i n i d e s . T h e p r e s e n t b o d y o f s y s t e m a t i c d a t a o n t h i s

m a s s r e g i o n i s u n a b l e , h o w e v e r , t o p e r m i t d i s t i n g u i s h i n g a m o n g t h e a l p h a

p a r t i c l e c l u s t e r , o c t u p o l e v i b r a t i o n a n d s t a t i c o c t u p o l e d e f o r m a t i o n

m o d e l s b e c a u s e o f t h e i r s i m i l a r i t i e s . H o w e v e r , e n o u g h o f t h i s i n f o r m a

t i o n i s now a v a i l a b l e t o s u g g e s t p o s s i b l e e x p e r i m e n t a l a p p r o a c h e s t o t h e

s o l u t i o n o f t h i s p r o b l e m . I n a d d i t i o n , o d d - A n u c l e i i n t h i s m a s s r e g i o n

a p p e a r t o b e h a v e v e r y m u ch l i k e n u c l e i o f s i m i l a r q u a d r u p o l e d e f o r m a t i o n

i n o t h e r r e g i o n s o f t h e p e r i o d i c t a b l e .

-22-

2. OVERVIEW OF THEORY

I n s e c t i o n s 2 . 1 - 2 . 4 we d i s c u s s t h e m o t i v a t i o n l e a d i n g t o a t t e m p t s

t o u n d e r s t a n d t h e o b s e r v a t i o n s o n t h e R a a n d T h i s o t o p e s w i t h i n t h e

f r a m e w o r k o f a l p h a p a r t i c l e c l u s t e r i n g , o c t u p o l e v i b r a t i o n a n d s t a t i c

o c t u p o l e d e f o r m a t i o n m o d e l s . H e r e we p r e s e n t a fe w o f t h e i r m o re i m p o r

t a n t i m p l i c a t i o n s f o r e v e n - e v e n n u c l e i . E a c h o f t h e s e m o d e l s a r i s e s i n

a n a t u r a l w a y f r o m b e h a v i o r o b s e r v e d n o t o n l y i n t h e a c t i n i d e r e g i o n b u t

a l s o t h r o u g h o u t t h e p e r i o d i c t a b l e . T h e e f f e c t s o f r e f l e c t i o n a s y m m e t r y

i n o d d - A n u c l e i a r e d i s c u s s e d i n s e c t i o n s 2 . 5 - 2 . 7 .

2 . 1 A L P H A P A R T I C L E C L U S T E R I N G I N H E A V Y N U C L E I A N D T H E

H Y B R I D M O D E L

T h e c o n c e p t o f a l p h a p a r t i c l e c l u s t e r i n g i n t h e l i g h t a c t i n i d e s i s

s u g g e s t e d b y tw o g e n e r a l o b s e r v a t i o n s . F i r s t , r e d u c e d w i d t h s f o r t h e

a l p h a p a r t i c l e d e c a y o f g r o u n d s t a t e s o f R a a n d T h i n t h e 2 1 8 < A < 2 3 0

i n t e r v a l a r e l a r g e . I n 2 1 8 R a , t h e g r o u n d s t a t e r e d u c e d a l p h a p a r t i c l e

w i d t h i s 2 0 % o f t h e W i g n e r L i m i t (T e 5 2 ) , a c o n v e n i e n t m e a s u r e o f m o l e c

u l a r c h a r a c t e r m o s t o f t e n u s e d i n c o n n e c t i o n w i t h h e a v y i o n s c a t t e r i n g .

I t i s g e n e r a l l y u n d e r s t o o d t h a t a s t a t e h a v i n g a r e d u c e d w i d t h f o r a

p a r t i c u l a r n u c l e a r f r a g m e n t o f 1 0 0 % o f t h e W i g n e r L i m i t i s a p u r e c o n

f i g u r a t i o n i n w h i c h t h e s a i d f r a g m e n t a n d t h e r e m a i n d e r o f t h e n u c l e u s

a r e t h e c o n s t i t u e n t s o f a p u r e m o l e c u l a r c o n f i g u r a t i o n i n w h i c h t h e c o n

s t i t u e n t s h a v e t a n g e n t s u r f a c e s . M o r e o v e r , a r e d u c e d w i d t h o f e v e n 2%

o f t h e W i g n e r L i m i t i s s t i l l c o n s i d e r e d t o b e " m o l e c u l a r " i n c o l l i s i o n s

o f h e a v y i o n s ( E b 8 1 ) .

T h e s e c o n d m o t i v a t i o n f o r t h i s m o d e l i s t h e w e l l e s t a b l i s h e d i m p o r

t a n c e o f a l p h a p a r t i c l e c l u s t e r i n g f o r l i g h t n u c l e i . We h a v e a l r e a d y

i n t r o d u c e d 180 a s o n e e x a m p l e o f a n u c l e u s i n w h i c h a l p h a p a r t i c l e c l u s

t e r i n g a p p e a r s t o p l a y a r o l e . I n f i g u r e 2 . 1 , we d i s p l a y a m u ch s t u d i e d

e x a m p l e , 20 N e . R e d u c e d a l p h a p a r t i c l e w i d t h s o f 2 0 % a n d 50% o f t h e W i g

n e r L i m i t f l a g t h o s e s t a t e s i n w h i c h t h e e f f e c t o f a l p h a p a r t i c l e c l u s

t e r i n g i s m o s t p r o n o u n c e d . A l t h o u g h t h e a r r a n g e m e n t o f s t a t e s i s n o t a s

s i m p l e a s t h a t o f t h e p r o p o s e d a l p h a p a r t i c l e c l u s t e r b a n d i n 180 , t h e

b a n d o f n e g a t i v e p a r i t y s t a t e s a s s o c i a t e d w i t h t h e r e f l e c t i o n a s y m m e t r i c

n a t u r e o f a l p h a p a r t i c l e c l u s t e r i n g ( J ^ l " , 3 ” , 5 " , 7 " w i t h <0 2 > = 5 0 % )ctc a n b e e a s i l y i d e n t i f i e d .

I n d e e d , e v e n t h e g r o u n d s t a t e o f 2 0 N e , w i t h a r e d u c e d w i d t h o f o n l y

5% o f t h e W i g n e r L i m i t , i s c o n s i d e r e d m o l e c u l a r i n t h e s e n s e t h a t i t s

w a v e f u n c t i o n c a n b e w r i t t e n a s

Y = *0*i6oXa_i60(ra_i60)'

w h e r e a n d $ 16q r e p r e s e n t t h e i n t r i n s i c s t a t e s o f t h e a l p h a p a r t i c l e

a n d 160 c o r e , r e s p e c t i v e l y , a n d x _ i 6q c h a r a c t e r i z e s t h e i r r e l a t i v e

p o s i t i o n . T h e d i f f e r e n c e i s t h a t t h o s e b a n d s w i t h r e d u c e d w i d t h s o f 4 0 %

a n d 50% o f t h e W i g n e r L i m i t h a v e r e l a t i v e l y w e l l s e p a r a t e d c l u s t e r s ,

w h e r e a s i n t h e g r o u n d s t a t e t h e y a c t u a l l y i n t e r p e n e t r a t e t o som e d e g r e e .

One e s s e n t i a l d i f f e r e n c e b e t w e e n 180 a n d 20 N e m u s t b e m e n t i o n e d ,

h o w e v e r . I n 180 , t h e a l p h a p a r t i c l e c l u s t e r c o n f i g u r a t i o n g i v e s r i s e t o

a n e l e c t r i c d i p o l e m o m e n t . T h i s i s n o t t h e c a s e i n 20 N e , s i n c e a + 160 i s

-24-

F i g u r e 2 . 1 P a r t i a l l e v e l s p e c t r u m

o f 2 0 Ne ( A j 8 3 ) .

-25-+00 +CO

+00 +CO

+V

+CO

+V

Iin

+ + +OJ O o'O J+ fO H

O A h" wo* ^

+ +oj o= o +^ m° A ii A*= oj a05= V + +

z o'+ 0 J oO O Jii ii

f= Ao j a05V

oco+a

= v°oi oo inII Mk A

cm az 05V

<02>

=5%

20 Ne

a s e l f - c o n j u g a t e s y s t e m s o t h a t E l t r a n s i t i o n s v a n i s h i n l e a d i n g o r d e r .

We s h o u l d e m p h a s i z e t h a t w h i l e 4H e + 14C s t a t e s p l a y a n e s s e n t i a l

r o l e i n 180 t h e y a r e n e v e r t h e l e s s h e a v i l y m i x e d w i t h 160 + 2 n a n d

2 0 N e + 2 ( p r o t o n h o l e ) c o n f i g u r a t i o n s a s w e l l a s t h e q u a d r u p o l e d e f o r m e d

s t a t e s o f 180 . C o n s e q u e n t l y , t h i s c o n f i g u r a t i o n m i x i n g m u s t b e t a k e n

i n t o a c c o u n t w hen i n t e r p r e t i n g t h e s p e c t r o s c o p i c i n f o r m a t i o n f o r t h i s

n u c l e u s .

A s s h o w n i n f i g u r e 1 . 4 , t h e s u g g e s t i o n t h a t l a r g e r e d u c e d a l p h a

p a r t i c l e w i d t h s a r e s i g n a t u r e s f o r a l p h a c l u s t e r i n g i n R a a n d T h a p p l i e s

t o c e r t a i n n u c l e i i n t h e A = 5 0 a n d 150 r e g i o n s a s w e l l .

A s m e n t i o n e d i n t h e i n t r o d u c t i o n , t h e S G A a p p r o a c h p r o v i d e s a u s e

f u l p h e n o m e n o l o g i c a l f r a m e w o r k f o r t h e a n a l y s i s o f d a t a o n q u a d r u p o l e

c o l l e c t i v i t y a n d m o l e c u l a r b e h a v i o r . I n t h e I n t e r a c t i n g B o s o n A p p r o x i

m a t i o n ( I B A ) , t h e S G A m o d e l f o r q u a d r u p o l e c o l l e c t i v i t y i n e v e n - e v e n

h e a v y n u c l e i , t h e n u c l e o n s o u t s i d e o f c l o s e d s h e l l s a r e p a i r e d , t h e

p a i r s b e i n g t r e a t e d a s b o s o n s . E a c h b o s o n c a n o c c u p y tw o q u a n t u m

s t a t e s : t h e g r o u n d , o r s - b o s o n , s t a t e w i t h a n g u l a r m omentum z e r o ; a n d

t h e e x c i t e d , o r d - b o s o n , s t a t e w i t h a n g u l a r m om entum o f 2FT a n d e x c i t a

t i o n e n e r g y e ^ . I n a d d i t i o n , t h e d - b o s o n c a n o c c u p y a n y o f t h e f i v e

m a g n e t i c s u b s t a t e s o f a s p i n 2 p a r t i c l e . C o n s e q u e n t l y , t h e r e a r e s i x

t y p e s o f b o s o n s p r e s e n t i n t h i s m o d e l . T h e t o t a l b o s o n n u m b e r ,

N = n + n , , s d

(w h e r e n a n d n , a r e t h e n u m b e r s o f s - a n d d - b o s o n s , r e s p e c t i v e l y ) i s s d

h e l d c o n s t a n t a t a v a l u e e q u a l t o h a l f t h e n u m b e r o f v a l e n c e n u c l e o n s

-26-

(the total of valence protons and neutrons). For example, 220Ra has 6

F i g u r e 2 . 2 C l a s s i c a l r e p r e s e n t a

t i o n o f q u a d r u p o l e a n d d i n u c l e a r

m o l e c u l a r d e g r e e s o f f r e e d o m a n d t h e

c o r r e s p o n d i n g s y m m e t r y g r o u p s U ( n )

( E n 8 4 ) .

YNT 70

(a) (b)z ii Z |i

//// ]/ IMA 1

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v a l e n c e p r o t o n s o v e r t h e s h e l l c l o s u r e Z = 8 2 , a n d 6 v a l e n c e n e u t r o n s o v e r

t h e m a g i c n u m b e r 1 2 6 , y i e l d i n g N = 6 . W hen t h e c r e a t i o n a n d a n n i h i l a t i o n

o p e r a t o r s f o r t h e s e b o s o n s a r e u s e d t o c o n s t r u c t a b o s o n n u m b e r - c o n s e r v

i n g h a m i l t o n i a n , t h e y a c t a s g e n e r a t o r s f o r t h e s y m m e t r y g r o u p o f t h e

q u a d r u p o l e c o l l e c t i v e s y s t e m , U ( 6 ) . A s i m p l y s t a t e d r u l e r e l a t i n g t h e

p o s s i b l e b o s o n s t a t e s ( i n c l u d i n g m a g n e t i c s u b s t a t e s ) t o t h e s y m m e t r y

g r o u p f o r t h e s y s t e m i s t h a t i f t h e r e a r e n b o s o n s t a t e s p r e s e n t , t h e n

t h e s y m m e t r y g r o u p i s U ( n ) .

T h e e i g e n s t a t e s o f a S p e c t r u m G e n e r a t i n g A l g e b r a c a n b e d e s c r i b e d

i n t e r m s o f a s e t o f p h y s i c a l b e n c h m a r k s c a l l e d d y n a m i c a l l i m i t s . I n

t h e s t u d y o f n u c l e a r c o l l e c t i v e p h e n o m e n a , a s u b g r o u p A o f t h e S G A s y m

m e t r y g r o u p i s a p h y s i c a l l y m e a n i n g f u l d y n a m i c a l l i m i t o f t h e S G A i f a n d

o n l y i f t h e s y m m e t r y g r o u p f o r a n g u l a r m om entum , 0 ( 3 ) , i s a s u b g r o u p o f

A ( l a 8 0 ) . O n l y t h e n d o t h e m o d e l s t a t e s h a v e a n g u l a r m om entum a s a

g o o d q u a n t u m n u m b e r . P h y s i c a l l y , a d y n a m i c a l l i m i t i s a l i m i t i n g c a s e

o f t h e p h y s i c a l m o t i o n d e s c r i b e d . F o r i n s t a n c e , t h e I B A h a s t h r e e

d y n a m i c a l l i m i t s : S U ( 3 ) , t h e d e f o r m e d s y m m e t r i c r o t o r ; U ( 5 ) , t h e a n h a r -

m o n i c v i b r a t o r ; a n d 0 ( 6 ) , t h e a x i a l l y a s y m m e t r i c , g a m m a - u n s t a b l e r o t o r .

A s i m p l i f i e d v e r s i o n o f t h e I B A h a m i l t o n i a n c a n i l l u s t r a t e how t h i s

m o d e l t r e a t s t h e q u a d r u p o l e v i b r a t i o n a l t o d e f o r m e d ( S U ( 5 ) t o S U ( 3 ) , i n

t h e l a n g u a g e o f t h e I B A ) t r a n s i t i o n . T h i s h a m i l t o n i a n i s

H = Ednd - KdQd'Qd. (2.2.1)

w h e r e n ^ a n d a r e d e f i n e d a s a b o v e , r e p r e s e n t s t h e q u a d r u p o l e -

q u a d r u p o l e i n t e r a c t i o n b e t w e e n b o s o n s , a n d i s a p a r a m e t e r r e p r e s e n t

i n g t h e s t r e n g t h o f t h i s t e r m . I n a d d i t i o n , i s t r e a t e d a s a p a r a m e -

-28-

S

t e r . When £ cj> > K d ' t 5 e e x c i t a t 4 o n s p e c t r u m c o n s i s t s o f d e g e n e r a t e

m u l t i p l e t s o f s t a t e s ( e a c h m u l t i p l e t c o r r e s p o n d i n g t o a g i v e n v a l u e o f

n ^ ) w h i c h a r e s p a c e d a p a r t . T h i s i s t h e b e h a v i o r we e x p e c t i n v i b r a

t i o n a l n u c l e i , a n d c o r r e s p o n d s t o t h e U ( 5 ) d y n a m i c a l l i m i t . We c a n

b r e a k t h e d e g e n e r a c i e s o f t h e m u l t i p l e t s b y i n c l u d i n g a t e r m

-29-

[k' + (3/8)Kd]Ld-L^ {2 2 2)

w h e r e L d g i v e s t h e a n g u l a r m omentum o f a b o s o n , i n t h e h a m i l t o n i a n .

T h e o p e r a t o r Q d c a n b e w r i t t e n i n t e r m s o f b o s o n c r e a t i o n ( s + a n d

d + ) a n d a n n i h i l a t i o n ( s a n d d ) o p e r a t o r s a s

Qd = ( s + d + d + s ) + ( l / 5 ) 1 / 2 x ( d + d ) ( 2 ) ,

w h e r e t h e s u p e r s c r i p t ( 2 ) d e n o t e s a c o u p l i n g t o tw o u n i t s o f a n g u l a r

m om entum , a n d x i s a p a r a m e t e r . I f < cj> > £ d an<4 X = 0 , t h e n t h e h a m i l t o n i a n

c o r r e s p o n d s t o t h e 0 ( 6 ) g a m m a - u n s t a b l e l i m i t ; a v a l u e x = ~ 2 . 9 5 8 y i e l d s

t h e S U ( 3 ) d e f o r m e d r o t o r l i m i t . T h e v i b r a t i o n a l t o d e f o r m e d t r a n s i t i o n

i s r e p r o d u c e d b y i n c r e a s i n g t h e i m p o r t a n c e o f t h e t e r m r e l a t i v e t o

t h e £ d n d t e r m ( i . e . b y i n c r e a s i n g

T h i s a l g e b r a i c p r e s c r i p t i o n c a n a l s o b e a p p l i e d t o s i m p l e m o l e c u l a r

b e h a v i o r . T h e r e s u l t , c a l l e d t h e v i b r o n m o d e l , u s e s s - b o s o n s ( 1 = 0 ) a n d

p - b o s o n s ( 1 = 1 ) , a n d h a s t h e s y m m e t r y g r o u p U ( 4 ) . N o b o s o n c o u n t i n g

r u l e s , s u c h a s t h o s e u s e d f o r t h e I B A , h a v e b e e n d e d u c e d f o r t h e v i b r o n

m o d e l . T h i s m o d e l h a s tw o d y n a m i c a l l i m i t s : 0 ( 4 ) , t h e r i g i d r o t a t i n g

m o l e c u l e w i t h s t i f f p a r t i c i p a n t n u c l e i ; a n d U ( 3 ) , a m o re c o m p l e x mode

w h i c h i s p r i m a r i l y o f o s c i l l a t o r y c h a r a c t e r ( l a 8 2 a ) . T h e s e tw o l i m i t s

r e s u l t i n e n e r g y l e v e l s p e c t r a a n d s e l e c t i o n r u l e s f o r g a m m a - r a y t r a n

s i t i o n s t h a t a r e i l l u s t r a t e d i n f i g u r e 2 . 3 . T h e s i m p l e v i b r o n m o d e l h a s

b e e n a p p l i e d t o a v a r i e t y o f n u c l e a r s y s t e m s , i n c l u d i n g 12C + 12 C ( E r 8 1 ,

C s 8 5 ) a n d i 2C + 160 (R u 8 4 ) .

I n t h e p r e s e n t s t u d y , we w i l l b e i n t e r e s t e d o n l y i n t h e U ( 3 ) l i m i t

o f t h e U ( 4 ) s y m m e t r y g r o u p ; t h e h a m i l t o n i a n f o r t h i s l i m i t i s

H = s n + a n ( n - 1 ) + k ' L * L , ( 2 . 2 . 3 )P P P P P P P P

w h e r e i s a p a r a m e t e r r e p r e s e n t i n g t h e e n e r g y r e q u i r e d t o p r o m o t e a

b o s o n f r o m a n s s t a t e t o a p s t a t e , a n d k 1 a n d a a r e t h e s t r e n g t hP

p a r a m e t e r s f o r t h e i r r e s p e c t i v e t e r m s . T h e p h e n o m e n o l o g y o f t h i s m o d e l

i s n o t a s w e l l u n d e r s t o o d a s t h a t o f t h e I B A ; h o w e v e r , we w o u l d e x p e c t

t h a t e w o u l d c o n t r o l t h e e x c i t a t i o n e n e r q y o f t h e b a n d h e a d o f t h e P

" d i p o l e v i b r a t i o n " b a n d , t h e b a n d o f s t a t e s h a v i n g n = 1 . F u r t h e r , t h eP

L ‘ L t e r m c l e a r l y a c t s t o b r e a k v i b r a t i o n a l m u l t i p l e t s .P P

When t h e i n t r i n s i c b e h a v i o r o f o n e o r b o t h o f t h e m o l e c u l a r c o n

s t i t u e n t s i s i m p o r t a n t , t h e s i t u a t i o n b e c o m e s m o r e c o m p l e x . F o r t h e

c a s e o f a l p h a p a r t i c l e c l u s t e r i n g i n h e a v y n u c l e i , t h e q u a d r u p o l e c o l

l e c t i v e e x c i t a t i o n s p e c t r u m o f t h e c o r e m u s t b e t a k e n i n t o a c c o u n t .

T h i s i s d o n e i n t h e H y b r i d M o d e l , a m a r r i a g e b e t w e e n t h e I B A a n d t h e

V i b r o n M o d e l (D a 8 3 ) . Two s e t s o f b o s o n s a r e u s e d , t h e t o t a l b o s o n num

b e r i n e a c h s e t b e i n g c o n s e r v e d . T h e f i r s t s e t c o n s i s t s o f t h e s a n d d

b o s o n s u s e d t o a c c o u n t f o r t h e q u a d r u p o l e m o t i o n o f t h e c o r e ; t h e m o l e c

u l a r a s p e c t s o f t h e c o r e - c l u s t e r s t r u c t u r e a r e d e s c r i b e d b y t h e s a n d p

b o s o n s o f t h e U ( 4 ) S G A . F o r c l a r i t y , we d e n o t e t h e b o s o n s o f t h e l a t t e r

s e t b y s * a n d p * . We a l s o u s e t h e n o t a t i o n

F i g u r e 2 . 3 A s c h e m a t i c r e p r e s e n t a

t i o n o f t h e e n e r g y s p e c t r a f o r t h e

tw o d y n a m i c a l l i m i t s o f t h e V i b r o n

M o d e l . T h e a r r o w s r e p r e s e n t f i r s t

o r d e r a l l o w e d E l t r a n s i t i o n s ( E n 8 4 ) .

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f o r t h e c o n s e r v e d b o s o n n u m b e r s f o r e a c h s e t . T h e s y m m e t r y g r o u p f o r

t h i s m o d e l i s s i m p l y U ( 6 ) $ U ( 4 ) . B o s o n s f r o m t h e tw o s e t s i n t e r a c t v i a

t h e h a m i l t o n i a n

% - >=«d'8p + 2* v y <2-2-4)

I t h a s b e e n f o u n d i n p r e v i o u s s t u d i e s u s i n g t h i s m o d e l t h a t t h e

U ( 3 ) v i b r o n h a m i l t o n i a n i s m o s t a p p r o p r i a t e f o r t h e d e s c r i p t i o n o f R a

a n d T h i s o t o p e s . C o n s e q u e n t l y , t h e h a m i l t o n i a n t h a t we u s e i n o u r s t u d y

i s t h e sum o f t h o s e f o u n d i n ( 2 . 2 . 1 ) , ( 2 . 2 . 2 ) , ( 2 . 2 . 3 ) a n d ( 2 . 2 . 4 ) .

T h e H y b r i d M o d e l c a l c u l a t i o n i t s e l f a s s u m e s t h e p r e s e n c e o f tw o

c o n f i g u r a t i o n s , o n e w i t h o u t a l p h a p a r t i c l e c l u s t e r i n g ( t h e z e r o a l p h a

c o n f i g u r a t i o n ) , f o r w h i c h N =0 a n d N , i s d e t e r m i n e d b y t h e u s u a l I B AP Qc o u n t i n g r u l e ; a n d a n o t h e r w i t h a s i n g l e v a l e n c e a l p h a p a r t i c l e c l u s t e r

( t h e o n e a l p h a c o n f i g u r a t i o n ) , i n w h i c h N ^ = 2 a n d i s tw o l e s s t h a n i n

t h e z e r o a l p h a c o n f i g u r a t i o n . T h e z e r o a l p h a c o n f i g u r a t i o n h a s o n l y t h e

q u a d r u p o l e d e g r e e o f f r e e d o m ; c o n s e q u e n t l y , o n l y p o s i t i v e p a r i t y e v e n

s p i n s t a t e s a r e p r e s e n t i n t h i s c o n f i g u r a t i o n . B o t h p o s i t i v e a n d n e g

a t i v e p a r i t y s t a t e s a r i s e f r o m t h e r e f l e c t i o n a s y m m e t r i c one a l p h a c o n

f i g u r a t i o n . S t a t e s f r o m b o t h c o n f i g u r a t i o n s a r e p u t i n t o a s i n g l e

s p a c e , i n w h i c h t h e y a r e m i x e d t h r o u g h t h e u s e o f a m i x i n g h a m i l t o n i a n

a n d f i r s t o r d e r p e r t u r b a t i o n t h e o r y .

V . = y ( s + 2 s * 2 + s * + 2 s 2 ) ,m ix

( w h e r e y i s a p a r a m e t e r ) a n d f i r s t o r d e r p e r t u r b a t i o n t h e o r y . T h e

e n e r g y d e n o m i n a t o r u s e d i n t h e e x p r e s s i o n f o r t h e p e r t u r b e d w a v e f u n c t i o n

-32-

i s a l s o a p a r a m e t e r , <f> . C o n f i g u r a t i o n m i x i n g t a k e s p l a c e a m o n g t h eap o s i t i v e p a r i t y s t a t e s , b u t t h e o n e a l p h a n e g a t i v e p a r i t y s t a t e s h a v e n o

z e r o a l p h a s t a t e s w i t h w h i c h t o i n t e r a c t a n d , t h u s , r e m a i n p u r e . An

e x a m p le i s s k e t c h e d i n f i g u r e 2 . 4 .

T h e H y b r i d M o d e l h a s b e e n a p p l i e d t o n u c l e i i n t h e r a r e e a r t h ,

l i g h t a c t i n i d e a n d h e a v y a c t i n i d e r e g i o n s ( D a 8 3 , D a 8 4 , D a 8 6 a , Da

8 6 b ) . I n p a r t i c u l a r , tw o s e p a r a t e s e t s o f p a r a m e t e r s h a v e b e e n u s e d t o

d e s c r i b e R a a n d T h i s o t o p e s ( D a 8 3 , D a 8 4 ) .

2 . 2 O C T U P O L E V I B R A T I O N A L B E H A V I O R I N E V E N - E V E N N U C L E I

I n t h e i n t r o d u c t i o n - we b r i e f l y d i s c u s s e d how t h e c o u p l i n g o f a n

o c t u p o l e p h o n o n t o t h e g r o u n d s t a t e q u a d r u p o l e c o l l e c t i v e b a n d c a n

a c c o u n t f o r t h e p r e s e n c e o f l o w - l y i n g n e g a t i v e p a r i t y s t a t e s . I t

a p p e a r s t h a t s u c h a c o u p l i n g p r o d u c e s a n a l t e r n a t i n g p a r i t y b a n d a t

s p i n s o f 3H a n d a b o v e i n t h e s p h e r i c a l n u c l e u s 1 5 0 G d . F u r t h e r , i n t h e

d e f o r m e d r a r e e a r t h n u c l e i t h e l o w e s t o c t u p o l e v i b r a t i o n a l s t a t e o f t e n

a r i s e s f r o m t h e c o u p l i n g o f a n o c t u p o l e p h o n o n w h o s e a n g u l a r m om entum

v e c t o r l i e s , i n t h e i n t r i n s i c f r a m e , n o r m a l t o t h e s y m m e t r y a x i s o f t h e

n u c l e u s . T h e a n g u l a r m om entum p r o j e c t i o n o n t h e s y m m e t r y a x i s i s a g o o d

q u a n t u m n u m b e r w h i c h i s d e n o t e d b y K . C o n s e q u e n t l y , t h e o c t u p o l e p h o n o n

we h a v e j u s t d e s c r i b e d h a s K ^ O " . T h i s a l i g n e d o c t u p o l e p h o n o n c a n g i v e

t h e s p i n - p a r i t y s e q u e n c e 1 " , 3 " , 5 " , e t c . , s u c h a s t h a t i n 1 62E r ( f i g u r e

2 . 5 ) .

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o n e a l p h a p a r t i c l e ( E n 8 4 ) .

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A . W . W R I G H T N U C L E A R S T R U C T U R E L A B O R A T O R Y - Y A L E

F i g u r e 2 . 5 A p a r t i a l l e v e l s p e c

t r u m o f 1 6 2 E r (H e 8 5 ) . T h e n e g a t i v e

p a r i t y s t a t e s s h o w n h a v e b e e n i n t e r

p r e t e d i n (H e 8 5 ) t o b e t h e r e s u l t o f

t h e c o u p l i n g o f a n o c t u p o l e p h o n o n

h a v i n g K ^ O " t o t h e q u a d r u p o l e

d e f o r m e d c o r e .

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i t y s t a t e s i n t h e t r a n s i t i o n a l r a r e - e a r t h n u c l e i i s s u p p o r t e d i n s e v e r a l

w a y s . F i r s t , s t r o n g E 3 t r a n s i t i o n s d e e x c i t e t h e l o w e s t 3 ' s t a t e s i n

t h e s e n u c l e i ( B e 7 1 ) . N e x t , tw o o c t u p o l e p h o n o n s t a t e s h a v e b e e n

o b s e r v e d i n 147G d a n d 1 4 8 G d ( K 1 8 2 , L u 8 4 ) ; t h e s e s t a t e s w e re r e c o g n i z e d

p r i m a r i l y b e c a u s e o f t h e i r d e e x c i t a t i o n b y a c a s c a d e o f tw o E 3 t r a n

s i t i o n s .

A n i n t e r p r e t a t i o n o f t h e c o r r e s p o n d i n g n e g a t i v e p a r i t y b a n d s o f

t h e s e n u c l e i i n t e r m s o f t h e c o u p l i n g o f a n o c t u p o l e p h o n o n t o s u c c e s

s i v e g r o u n d s t a t e b a n d m e m b e r s i s s u g g e s t e d b y t h e s i m i l a r i t y o f t h e

s t r u c t u r e s o f t h e g r o u n d s t a t e a n d n e g a t i v e p a r i t y b a n d s a s w e l l a s t h e

e n h a n c e d E l t r a n s i t i o n s c o n n e c t i n g t h e tw o b a n d s . F u r t h e r , a n u m b e r o f

m o d e l s i n c o r p o r a t i n g o c t u p o l e p h o n o n s h a v e s u c c e s s f u l l y r e p r o d u c e d

o b s e r v e d e x c i t a t i o n s p e c t r a ( s e e , f o r i n s t a n c e , H a 77 a n d S u 7 7 ) .

We s h o u l d a l s o n o t e t h e r e s u l t s o f ( V o 7 6 ) , w h i c h s u g g e s t t h a t t h e

c h a r a c t e r o f n e g a t i v e p a r i t y s t a t e s i n d e f o r m e d r a r e e a r t h n u c l e i

c h a n g e s f r o m o c t u p o l e v i b r a t i o n a l t o tw o q u a s i p a r t i c l e a s t h e s p i n

i n c r e a s e s f r o m 9 R t o 15 f i . S e v e r a l a u t h o r s (H a 7 7 , H a 7 9 , S u 7 7 ) h a v e

n o t e d t h a t t h i s s u g g e s t i o n m ay a p p l y t o t r a n s i t i o n a l l a n t h a n i d e n u c l e i

a s w e l l . F u r t h e r , i t i s a l s o p r e d i c t e d i n ( V o 7 6 ) t h a t a n e q u i v a l e n t

t r a n s f o r m a t i o n o c c u r s i n t h e h e a v y a c t i n i d e s ( f o r e x a m p l e , 2 3 8 U ) a t

s p i n s n e a r 2 5 R .

A n o c t u p o l e v i b r a t i o n a l b e h a v i o r c a n p r o d u c e E l t r a n s i t i o n s c o n s i d

e r a b l y s t r o n g e r t h a n t h o s e a r i s i n g f r o m s i n g l e p a r t i c l e t r a n s i t i o n s . B y

s i m p l y c o n s i d e r i n g t h e e x p r e s s i o n f o r t h e d i p o l e m om ent r e s u l t i n g f r o m

a n o c t u p o l e s h a p e , a n e x p r e s s i o n w h i c h i s common t o b o t h t h e S t r u t i n s k i

( S t 5 7 ) a n d B o h r - M o t t e l s o n ( B o 5 7 ) t r e a t m e n t s d i s c u s s e d e a r l i e r , n a m e l y :

-36-

-37-

a n d t h e e x p r e s s i o n f o r t h e f i r s t o r d e r r e d u c e d E l m a t r i x e l e m e n t i n a

q u a d r u p o l e d e f o r m e d n u c l e u s , n a m e l y :

B ( E l : I . - > I tf) = ( 3 / 4 -it) < D 2>< I . 1 0 0 | I . 1 I . 0> ( 2 . 2 . 2 )i f l i f

we s e e t h a t

B ( E 1 : I . + I f ) cc 02 2 < 8 3 2 > ( 2 . 2 . 3 )

T h u s , a n o c t u p o l e p h o n o n i s s u f f i c i e n t t o g i v e a s t r o n g E l t r a n s i t i o n

w he n e v e n a s m a l l q u a d r u p o l e d e f o r m a t i o n e x i s t s . I t b e a r s e m p h a s i s ,

h o w e v e r , t h a t w h i l e r e d u c e d m a t r i x e l e m e n t s o f E l t r a n s i t i o n s a s s o c i a t e d

w i t h o c t u p o l e v i b r a t i o n s i n t h e l a n t h a n i d e r e g i o n a r e e n h a n c e d r e l a t i v e

t o s i n g l e p a r t i c l e E l d e e x c i t a t i o n s i n h e a v y n u c l e i , t h e y a r e s t i l l a

f a c t o r o f t e n l o w e r t h a n t h o s e m e a s u r e d i n 2 1 8 R a (G a 8 6 ) .

O ne a p p a r e n t s h o r t c o m i n g i n t h e s e m o d e l s , h o w e v e r , c o n c e r n s t h e

p r e d i c t e d d e p e n d e n c e o f t h e s t r e n g t h o f t h e E l t r a n s i t i o n o n t h e p a r i t y

o f t h e i n i t i a l s t a t e . S e v e r a l c a l c u l a t i o n s , i n c l u d i n g t h o s e i n ( S u 7 7 )

a n d (H a 7 7 ) , p r e d i c t t h a t t h e E l t r a n s i t i o n s o r i g i n a t i n g f r o m p o s i t i v e

p a r i t y s t a t e s s h o u l d b e s i g n i f i c a n t l y l o w e r t h a n t h o s e o r i g i n a t i n g f r o m

n e g a t i v e p a r i t y s t a t e s i n s p h e r i c a l a n d t r a n s i t i o n a l n u c l e i . A d i f f e r

e n c e o f a f a c t o r o f t h r e e i s p r e d i c t e d i n (H a 7 9 ) f o r 1 4 8 Sm; a c a l c u l a

t i o n i n (H a 7 7 ) f o r 1 50G d u s i n g a d i f f e r e n t t e c h n i q u e p r e d i c t s a f a c t o r

o f 1 0 0 . H o w e v e r , m o s t n u c l e i i n t h i s n e i g h b o r h o o d s h o w n o e v i d e n c e f o r

s u c h a d i f f e r e n c e ( s e e f i g u r e 2 . 6 ) . T h e c a l c u l a t i o n i n ( S u 7 7 ) o v e r

c o m e s t h i s d i f f i c u l t y b y i n c l u d i n g a n a d d i t i o n a l t w o - b o d y t e r m i n t h e

D = cAZep p (2.2.1)

F i g u r e 2 . 6 T h e r a t i o o f th e

r e d u c e d m a t r i x e l e m e n t s o f t h e E l a n d

E 2 d e e x c i t a t i o n s , d e n o t e d b y

B ( E 1 : J - » J - 1 ) / B ( E 2 : J - + J - 2 ) , b e t w e e n

s t a t e s b e l o n g i n g t o t h e g r o u n d s t a t e

a n d o c t u p o l e b a n d s i n 150G d , 148Sm

a n d 1 5 0 Sm. N o o d d - e v e n p a r i t y s t a g

g e r i n g o f l a r g e m a g n i t u d e i s e v i d e n t .

D a t a a r e t a k e n f r o m r e f e r e n c e s ( S u

7 7 , H a 7 7 , P e 8 4 b ) .

B(E

l: J

—J

-D/B

(E2

: J

-J-2

) (W

.u.)

-38-

T h e p r i m a r y o b j e c t i o n t o t h e i n t e r p r e t a t i o n o f R a a n d T h b e h a v i o r

i n t e r m s o f o c t u p o l e v i b r a t i o n s h a s b e e n t h a t tw o o c t u p o l e p h o n o n s t a t e s

o f s p i n - p a r i t y 0 + ( w h i c h i n t h e c a s e o f h a r m o n i c o s c i l l a t i o n w o u l d b e

f o u n d a t t w i c e t h e e x c i t a t i o n e n e r g y o f t h e n e g a t i v e p a r i t y b a n d h e a d )

h a v e n o t b e e n o b s e r v e d i n e v e n - e v e n R a a n d T h i s o t o p e s n e a r m a s s 2 2 5 ,

w h e r e e x t e n s i v e s t u d i e s o f s t a t e s o f l o w s p i n a n d e x c i t a t i o n e n e r g y h a v e

b e e n p e r f o r m e d t h r o u g h e x a m i n a t i o n o f a l p h a - a n d b e t a - p a r t i c l e d e c a y .

T h i s a r g u m e n t c a n b e a n s w e r e d i n tw o w a y s . F i r s t , P i e p e n b r i n g ( P i 8 2 )

h a s p e r f o r m e d c a l c u l a t i o n s t h a t sh o w t h a t a n h a r m o n i c i t i e s i n t h e v i b r a

t i o n a l b e h a v i o r o f R a a n d T h c a n d r i v e t h e t w o - p h o n o n s t a t e s t o m uch

h i g h e r e n e r g i e s , p o s s i b l y a s h i g h a s f i v e t i m e s t h e o n e - p h o n o n e n e r g y .

S e c o n d , i t i s c l e a r t h a t a n y s t a t i c o c t u p o l e d e f o r m e d b e h a v i o r m u s t b e

t h e e n d r e s u l t o f , i n a n a l o g y t o t h e q u a d r u p o l e c a s e , a s m o o t h t r a n

s i t i o n f r o m o c t u p o l e v i b r a t i o n t o o c t u p o l e d e f o r m a t i o n t a k i n g p l a c e o v e r

a s e r i e s o f n u c l e i . T h i s c o u l d m e a n , f o r i n s t a n c e , t h a t 2 2 4 R a h a s a

s t a t i c o c t u p o l e d e f o r m a t i o n , w h i l e 2 2 0 R a s i m p l y e x h i b i t s a n o c t u p o l e

v i b r a t i o n a l b e h a v i o r ( C h 7 9 ) .

2 . 3 S T A T I C O C T U P O L E D E F O R M A T I O N S I N E V E N - E V E N N U C L E I

J u s t a s s t a t i c a l l y d e f o r m e d q u a d r u p o l e s h a p e s e x i s t i n n u c l e i , we

m i g h t e x p e c t t h a t s t a t i c o c t u p o l e s h a p e s c a n b e f o u n d a s w e l l . U n l i k e

v i b r a t i o n a l b e h a v i o r , a s t a t i c s h a p e c o m p o n e n t p r e s e n t i n t h e g r o u n d

s t a t e a f f e c t s g r o u n d s t a t e p r o p e r t i e s , s u c h a s t h e n u c l e a r m a s s . I n

-39-

quantum operator for the El transition.

1 9 8 1 , M t f l l e r a n d N i x p u b l i s h e d a c a l c u l a t i o n o f n u c l e a r m a s s e s t a k i n g

b o t h q u a d r u p o l e a n d h e x a d e c a p o l e d e f o r m a t i o n s i n t o a c c o u n t (M o 8 1 b ) .

T h e i r c a l c u l a t i o n i s q u i t e s u c c e s s f u l o v e r t h e e n t i r e r a n g e o f h e a v y

( A > 8 0 ) n u c l e i ; h o w e v e r , r e l a t i v e l y l a r g e d i s c r e p a n c i e s a r e f o u n d b e t w e e n

t h e o r e t i c a l r e s u l t s a n d e x p e r i m e n t a l v a l u e s a r o u n d m a s s 2 2 0 ( s e e f i g u r e

2 . 7 ) . M o l l e r a n d N i x c o m m e n t e d t h a t t h i s p r o b l e m m ay a r i s e f r o m t h e

p r e s e n c e o f a s t a t i c o c t u p o l e c o m p o n e n t i n t h e g r o u n d s t a t e s h a p e o f R a

a n d T h i s o t o p e s (M o 8 1 a ) .

T h i s o b s e r v a t i o n i n s p i r e d s e v e r a l t h e o r e t i c a l i n v e s t i g a t i o n s o f

g r o u n d s t a t e p o t e n t i a l e n e r g y s u r f a c e s o f R a a n d T h . T h e m o s t p r o m i n e n t

o f t h e s e a r e s t u d i e s b y L e a n d e r , e t a l . , ( L e 8 2 a ) a n d b y N a z a r e w i c z , e t

a l . (N a 8 4 b ) . B o t h s t u d i e s i n c l u d e d q u a d r u p o l e , o c t u p o l e a n d h e x a d e c a

p o l e d e g r e e s o f f r e e d o m a n d c o n c l u d e d t h a t s t a b l e o c t u p o l e d e f o r m e d m i n

im a d o o c c u r i n t h e p o t e n t i a l e n e r g y s u r f a c e s f o r s e v e r a l R a a n d T h i s o

t o p e s . I n p a r t i c u l a r , t h e r e s u l t s o f N a z a r e w i c z , e t a l . ( s e e f i g u r e

2 . 8 ) p r e d i c t t h a t 2 2 2 ' 2 2 4 R a a n d 2 2 2 - 2 2 6 ^ ^ h a v e s t a t i c s h a p e s w h i c h

i n c l u d e b o t h q u a d r u p o l e a n d o c t u p o l e m o m e n t s . N a z a r e w i c z , e t a l . a l s o

e x t e n d e d t h e c a l c u l a t i o n s t o o t h e r r e g i o n s o f t h e p e r i o d i c t a b l e a n d

f o u n d t h a t h i s m e t h o d p r e d i c t e d a s h a l l o w s t a t i c o c t u p o l e d e f o r m e d m i n i

mum i n t h e p o t e n t i a l e n e r g y s u r f a c e o f t h e 1 4 6 B a g r o u n d s t a t e a s w e l l .

We s h o u l d n o t e h e r e t h a t a c o m p l e t e a n a l y s i s o f g r o u n d s t a t e p o t e n

t i a l e n e r g i e s w o u l d i n c l u d e a s f r e e p a r a m e t e r s a l a r g e n u m b e r o f m u l t i

p o l e s o f o r d e r h i g h e r t h a n h e x a d e c a p o l e a s w e l l a s t h e l o w e r o r d e r s

i n c l u d e d i n (N a 8 4 b ) . H o w e v e r , s u c h c a l c u l a t i o n s a r e i m p r a c t i c a l o n

c o n v e n t i o n a l c o m p u t i n g e q u i p m e n t . I n s t e a d , N a z a r e w i c z s h o w e d t h a t t h e

i n c l u s i o n o f a n d c o m p o n e n t s o f r e a s o n a b l e s i z e w o u l d a f f e c t t h e

-40-

F i g u r e 2 . 7 A c o m p a r i s o n o f g r o u n d

s t a t e m i c r o s c o p i c e n e r g i e s , a s c a l c u

l a t e d i n (M o 8 1 b ) , t o e x p e r i m e n t a l

v a l u e s f o r 1 3 2 3 n u c l i d e s (M o 8 1 b ) .

T h e b o t t o m l i n e s h o w s t h e d i f f e r e n c e

b e t w e e n e x p e r i m e n t a l a n d c a l c u l a t e d

v a l u e s . I s o t o p e s a r e c o n n e c t e d b y

l i n e s . T h e l a r g e d i s c r e p a n c y a r i s i n g

i n R a a n d T h i s l o c a t e d n e a r n e u t r o n

n u m b e r 1 3 4 .

Gr

ou

nd

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te

Mic

rosc

op

ic

En

erg

y

(Me

V) . i i i 1 m i i i [ i r i i | i i i i | i i m | t i i r | i i i i | i i t i | \ i i i | r i 'i r~p "i i i | r r i i | i i i i | i i i i | i i > \ |■Ilf!

10 i-

0 :

-10 r

0 :

-10 r

Experimental Microscopic zero-p o in t energies

Discrepancy (Expt. - Calc.)0

-10 \ i l 1 l Li I l l 1 i i i i 1 i » t ' 1 \ * i * I \ t * » I i * t \ I * i \ \ \ i \ i t 1 * I i ' I 1 i i * 1 * i l l 1 i t i i \ , t i , I0 20 40 60 80 100 120 140 160

Neutron Number N

p o t e n t i a l e n e r g y b y s m a l l a m o u n t s , l e s s t h a n .5 M e V , i n q u a d r u p o l e

s h a p e s r a n g i n g f r o m s p h e r i c a l t o e x t r e m e e l o n g a t i o n ( 3 2 = 1 . 0 ) . T h i s

r e s u l t l e n t c r e d i b i l i t y t o h i s d e l e t i o n o f h i g h e r m o m e n t s f r o m h i s c a l

c u l a t i o n .

A r o u g h d e s c r i p t i o n o f t h e e n e r g y s p e c t r a a r i s i n g f r o m d i f f e r e n t

f o r m s o f o c t u p o l e c o l l e c t i v i t y h a s b e e n g i v e n i n r e f e r e n c e ( L e 8 2 a ) . A s

i l l u s t r a t e d i n f i g u r e 2 . 8 , t h e p o t e n t i a l e n e r g y s u r f a c e s a r e s y m m e t r i c

a b o u t C o n s e q u e n t l y , w he n t h e p o t e n t i a l e n e r g y i s m i n i m i z e d w i t h

r e s p e c t t o 3 ^ anc 3 4 an<3 p l o t t e d a s a f u n c t i o n o f 3 ^ , t h e r e s u l t a p p e a r s

a s i n f i g u r e 2 . 9 . I f t h e p o t e n t i a l b a r r i e r b e t w e e n t h e tw o m i n i m a i s

i n f i n i t e , a p e r f e c t l y s t a t i c s h a p e r e s u l t s a n d t h e e n e r g y l e v e l s e q u e n c e

i s 0 + , 1 " , 2 + , 3 " , e t c . F o r a f i n i t e b a r r i e r , t u n n e l i n g b e t w e e n t h e m i n i m a

r e s u l t s i n t h e d i s p l a c e m e n t o f t h e n e g a t i v e - p a r i t y s t a t e s w i t h r e s p e c t

t o t h e p o s i t i v e - p a r i t y s t a t e s . F i n a l l y , i n t h e p u r e l y v i b r a t i o n a l s i t u

a t i o n t h e n e g a t i v e - p a r i t y s t a t e s a r e d i s p l a c e d r e l a t i v e t o t h e g r o u n d

s t a t e b y t h e o n e p h o n o n e n e r g y , Tiw-j. T h e d e f o r m e d t o v i b r a t i o n a l t r a n

s i t i o n d e p i c t e d h e r e i s p r e c i s e l y t h a t t o w h i c h we r e f e r r e d i n t h e p r e

v i o u s s e c t i o n .

T h e i d e a o f a s t a t i c o c t u p o l e s h a p e h a s a l s o b e e n a p p l i e d t o

e x p l a i n a n a p p a r e n t a n o m a l y i n t h e h i g h s p i n b e h a v i o r o f 2 2 2 T h . D u d e k ,

e t a l . ( D u 8 2 ) f o u n d t h a t t h e c r a n k i n g m o d e l b a s e d o n l y o n q u a d r u p o l e

a n d h e x a d e c a p o l e d e f o r m a t i o n s p r e d i c t e d a p r o n o u n c e d b a c k b e n d , a r o t a

t i o n a l s i g n a t u r e o f t h e b r e a k i n g o f a p a i r o f n u c l e o n s a t a p a r t i c u l a r

r o t a t i o n a l f r e q u e n c y ( b e c a u s e o f C o r i o l i s e f f e c t s ) , o n t h e y r a s t l i n e i n

2 2 2 T h ( s e e f i g u r e 2 . 1 0 ) . S u b s e q u e n t m e a s u r e m e n t s o f t h e e x c i t a t i o n

s p e c t r u m o f 2 2 2 T h w e r e n o t r e p r o d u c e d b y t h i s p r e d i c t i o n ( B o 8 5 , Wa 8 3 ) .

-42-

F i g u r e 2 . 8 P o t e n t i a l e n e r g y s u r

f a c e s f o r t h e g r o u n d s t a t e s o f 1 4 6 B a ,

2 2 0 R a , 2 2 2 R a a n d 2 2 4 R a a s p r e s e n t e d

i n ( N a 8 4 b ) . T h e e n e r g y i s m i n i m i z e d

w i t h r e s p e c t t o 8 4 a t e a c h p o i n t , a n d

t h e 8 5 a n d 8 6 p a r a m e t e r s a r e s e t a s

f u n c t i o n s o f 8 ^ / 8 3 a n i 3 8 4

e x p r e s s i o n s s t a t e d i n t h e r e f e r e n c e .

T h e c o n t o u r l i n e s a r e 0 . 1 M eV a p a r t .

T h e d a s h e d l i n e s i n t h e 2 2 0 R a s u r f a c e

t r a c e t h e p a t h o f m ax im um s o f t n e s s

( t h e m o s t l i k e l y p a t h f o r o s c i l l a

t i o n s ) i n t h e ( 8 2 - 8 3 ) p l a n e .

ICO<TI91*0

zsl80*0 0*0

:S/

'222

frEO 91*0 80*0 0*0 80*0-

F i g u r e 2 . 9 S c h e m a t i c i l l u s t r a t i o n

o f how a K=0 o c t u p o l e b a n d i s

e x p e c t e d t o d e p e n d o n t h e p o t e n t i a l

e n e r g y b a r r i e r ( L e 8 2 a ) . T h e p e r t u r

b a t i o n a p p r o a c h t o t h e s o l u t i o n o f a

d o u b l e o s c i l l a t o r i s u s e d . F o r a n

i n f i n i t e p o t e n t i a l b a r r i e r b e t w e e n

m i r r o r i m a g e s , t h e p o s i t i v e a n d n e g

a t i v e p a r i t y s t a t e s i n t h e e v e n - e v e n

n u c l e u s f o r m a n a l t e r n a t i n g p a r i t y

b a n d . A s t h e h e i g h t o f t h e b a r r i e r

i s r e d u c e d , t h e n e g a t i v e p a r i t y

s t a t e s a r e d i s p l a c e d h i g h e r i n

e n e r g y .

4+

3 +even-even ------- 2 1“

0 +

11/20 d d - A -------- 9/2

7 /2 1+1+

l+€ 3<0 ° € 3 >0 € 3< 0 ° € 3 >0

0+

11/2+'9 /2 +7 /2 +5 /2+

3 - . --------- 3

r4+ |-

2+ 0+ 0

2 phonon vib.

7 /2— i— 5 /2— !— 9 / 2 "

9 / 2 " ------ ------- H /2*----------7 / 2 “

■5/2“ ------- 9 /2+ 5 / 2 ~5 /2 ------- --------? /2 + 5 /2 *

+ +

F i g u r e 2 . 1 0 A n g u l a r m om entum,

I ( x ) = J + l / 2 , v s . Tiu = 1 / 2 ( E ( J + l ) -

E ( J - l ) ) c a l c u l a t e d f o r b a n d s i n

2 2 2 T h w i t h ( s o l i d c u r v e ) a n d w i t h o u t

( d a s h e d c u r v e s ) s t a t i c o c t u p o l e

d e f o r m a t i o n . S o l i d a n d o p e n p o i n t s

a r e m e a s u r e d v a l u e s f r o m p o s i t i v e a n d

n e g a t i v e p a r i t y b a n d s , r e s p e c t i v e l y

( N a 8 4 a ) . T h e s h a p e p a r a m e t e r s u s e d

i n t h e c a l c u l a t i o n f o r t h e s t a t i c

o c t u p o l e d e f o r m e d s h a p e w e r e s e t b y

t h e sam e p r o c e d u r e a s i l l u s t r a t e d i n

f i g u r e 2 . 8 ; t h e y a r e 32= 0 . 1 1 4 ,

83 = 0 . 0 9 6 , 34 = 0 . 0 6 7 8 , 3 5= 0 . 0 0 6 7 a n d

3 = 0 . 0 0 2 8 . I n t h e 3 o = 0 c a l c u l a t i o n , o 3a l l o t h e r p a r a m e t e r s w e r e i d e n t i c a l

t o t h o s e i n t h e o c t u p o l e d e f o r m e d

c a l c u l a t i o n .

1i cu ( M e V )

H o w e v e r , a new c a l c u l a t i o n u s i n g a c r a n k i n g m o d e l i n c l u d i n g t h e s t a t i c

o c t u p o l e s h a p e i n a d d i t i o n t o t h e q u a d r u p o l e a n d h e x a d e c a p o l e c o m p o n e n t s

w a s a b l e t o a c c o u n t f o r t h e a b s e n c e o f t h i s b a c k b e n d (N a 8 4 a , N a 8 5 ) .

F i n a l l y , i t i s c l e a r f r o m e q u a t i o n ( 2 . 2 . 3 ) t h a t t h e p r e s e n c e o f a

s t a t i c o c t u p o l e s h a p e c a n l e a d t o s t r o n g E l t r a n s i t i o n s . T h e r e f o r e , t h e

s t a t i c o c t u p o l e p i c t u r e c a n a c c o u n t f o r t h e a v a i l a b l e e x p e r i m e n t a l

i n f o r m a t i o n , a t l e a s t i n a q u a l i t a t i v e w a y .

A n S G A d e s i g n e d t o i n c o r p o r a t e t h e o c t u p o l e d e g r e e o f f r e e d o m h a s

r e c e n t l y b e e n d e v e l o p e d ( E n 8 5 ) . To t h i s p o i n t , o n l y p r e l i m i n a r y i n v e s

t i g a t i o n s o f R a a n d T h n u c l e i h a v e b e e n p e r f o r m e d u s i n g t h i s m o d e l ; c o n

s e q u e n t l y , i t i s n o t y e t c l e a r how h e l p f u l t h i s c a l c u l a t i o n a l t o o l w i l l

b e i n g a i n i n g a n u n d e r s t a n d i n g o f t h e i r b e h a v i o r .

2 . 4 A F E W C O M M E N T S O N T H E R E L A T I O N S H I P B E T W E E N T H E A L P H A

P A R T I C L E C L U S T E R A N D S T A T I C O C T U P O L E M O D E L S

B y m e r e l y l o o k i n g a t s k e t c h e s o f t h e s h a p e s i n v o l v e d ( s e e f i g u r e

1 . 4 ) , o n e c a n i n f e r t h a t t h e a l p h a p a r t i c l e c l u s t e r a n d s t a t i c o c t u p o l e

p i c t u r e s a r e r e l a t e d . We h a v e a l r e a d y n o t e d t h a t f r o m a p h e n o m e n o l o g i

c a l p o i n t o f v i e w , t h e a l p h a p a r t i c l e c l u s t e r , o c t u p o l e v i b r a t i o n , a n d

s t a t i c o c t u p o l e d e f o r m a t i o n m o d e l s a r e e q u i v a l e n t i n m a n y r e s p e c t s .

E a c h c a n a c c o u n t f o r o b s e r v e d e n e r g y s p e c t r a , a n d e a c h c a n l e a d t o

e n h a n c e d E l t r a n s i t i o n s . F u r t h e r , i t i s n o t c l e a r t h a t t h e e x i s t e n c e o f

a n i d e n t i f i a b l e v a l e n c e a l p h a p a r t i c l e i s r e q u i r e d i n o r d e r t o y i e l d a

l a r g e a l p h a p a r t i c l e r e d u c e d w i d t h ( a s , f o r e x a m p l e , i n t h e c a s e o f

-46-

T h e m o s t o b v i o u s g e o m e t r i c d i s t i n c t i o n b e t w e e n t h e a l p h a p a r t i c l e

c l u s t e r a n d t h e m o s t c o m m o n ly s u g g e s t e d p a r a m e t e r i z a t i o n s o f t h e o c t u

p o l e s h a p e i s t h a t t h e n a r r o w e n d o n t h e o c t u p o l e i s , f o r r e a s o n a b l e

v a l u e s o f 8 3 / c o n s i d e r a b l y l a r g e r t h a n t h e a l p h a p a r t i c l e ( s e e f i g u r e

1 . 4 ) . T h e e s s e n t i a l i n g r e d i e n t i n d i s p o s i n g o f t h e b a c k b e n d i n t h e

o c t u p o l e c r a n k i n g m o d e l ( N a 8 4 a , N a 8 5 ) i s t h e s t a t i c r e f l e c t i o n a s y m m e

t r y i n t h e s y s t e m . W h e t h e r o r n o t t h e d i f f e r e n c e i n t h e s i z e s o f t h e

n a r r o w e n d s o f t h e tw o s h a p e s w o u l d h a v e a s i g n i f i c a n t e f f e c t o n t h e

c r a n k i n g m o d e l c a l c u l a t i o n i s a s y e t a n u n a n s w e r e d q u e s t i o n . S i m i l a r

q u e s t i o n s c o u l d b e r a i s e d i n r e l a t i o n t o t h e p o t e n t i a l e n e r g y s u r f a c e

c a l c u l a t i o n s .

T h e i d e a t h a t o c t u p o l e a n d a l p h a p a r t i c l e c l u s t e r c o n f i g u r a t i o n s

m i g h t c o e x i s t i n l a n t h a n i d e n u c l e i w a s r e c e n t l y a d v a n c e d b y I a c h e l l o ( l a

8 5 ) . T h e r e i s n o c l e a r r e a s o n w hy t h i s p o s s i b i l i t y s h o u l d n o t a l s o b e

c o n s i d e r e d f o r a c t i n i d e n u c l e i .

2 . 5 W E A K A N D I N T E R M E D I A T E C O U P L I N G I N T R A N S I T I O N A L O D D - A

N U C L E I

We w i l l now m ove t o t h e q u e s t i o n o f w h e t h e r r e f l e c t i o n a s y m m e t r y

f u n d a m e n t a l l y a f f e c t s s i n g l e p a r t i c l e b e h a v i o r . T h e e m p h a s i s i n t h i s

s e c t i o n w i l l b e o n s p h e r i c a l o r n e a r - s p h e r i c a l n u c l e i . I n t h e n e x t s e c

t i o n we w i l l e x a m i n e t h e q u e s t i o n o f d e f o r m e d n u c l e i .

-47-

2 °Ne).

In general, the question of the coupling of the collective behavior

o f a n u c l e u s t o t h e d e g r e e s o f f r e e d o m o f a s i n g l e p a r t i c l e c a n be

s p e c i f i e d b y a H a m i l t o n i a n o f t h e f o r m

-48-

H = H + H + H . t ( 2 . 5 . 1 ) c o r e s p i n t

T h e t e r m H i s a n a p p r o p r i a t e c o l l e c t i v e H a m i l t o n i a n , H i s a s h e l l c o r e s p

m o d e l - l i k e t e r m f o r t h e s i n g l e p a r t i c l e , a n d r e p r e s e n t s a n i n t e r a c

t i o n b e t w e e n t h e c o r e a n d s i n g l e p a r t i c l e . T h e c o u p l i n g s t r e n g t h , a n d

t h u s t h e a p p r o a c h t h a t m u s t b e t a k e n t o t h e p r o b l e m , d e p e n d o n t h e r e l a

t i o n s h i p o f H . . t o H a n d Hm t c o r e s p

A f r e q u e n t l y u s e d f o r m f o r i s

f „ ( r ) ( 2 . 5 . 2 )m t £m £ £m c o r e £m s p

(D e 6 1 ) . A s o n e m i g h t e x p e c t , t h e q u a d r u p o l e c o m p o n e n t o f ( 2 . 5 . 2 ) i s

d o m i n a n t i n m o s t c a s e s . F o r t h e p u r p o s e s o f t h i s s e c t i o n , we w i l l c o n

s i d e r o n l y t h i s t e r m o f . T h i s t e r m a m o u n t s t o a p r o d u c t o f t h e

c o r e a n d s i n g l e p a r t i c l e q u a d r u p o l e m o m e n t s . T h e r e f o r e , we w o u l d e x p e c t

t h a t f o r a n e a r l y s p h e r i c a l n u c l e u s i s q u i t e s m a l l ' a n d t h a t i t v a n

i s h e s i d e n t i c a l l y i n t h e s p h e r i c a l c a s e . I f t h i s i s t h e c a s e , t h e n we

c a n t r e a t t h e e i g e n v e c t o r s o f H + H a s t h e u n p e r t u r b e d s o l u t i o n s3 c o r e s p

t o a p r o b l e m i n w h i c h i s t h e p e r t u r b a t i o n . We s t a r t w i t h t h e m o s t

g e n e r a l f o r m f o r a n e i g e n s t a t e o f H + H ,^ ^ c o r e s p

I J j J H > = I , _ ( J j J | m l m2 M )c o r e s p m l , m 2 c o r e s p

| J m l > | j m2 > . ( 2 . 5 . 3 )c o r e s p

S i n c e t h e s e s t a t e s c o r r e s p o n d t o H ^n t = 0 “ t4ie w e a k c o u p l i n g l i m i t - t h e y

a r e r e f e r r e d t o a s t h e w e a k c o u p l i n g b a s i s . I n t h e w e a k c o u p l i n g l i m i t ,

t h e e n e r g y l e v e l s p e c t r u m c o n s i s t s o f d e g e n e r a t e m u l t i p l e t s o f s t a t e s ,

e a c h m u l t i p l e t b e i n g f o u n d a t t h e e n e r g y o f i t s c o r r e s p o n d i n g c o r e

s t a t e . T h e m u l t i p l e t s h a v e s t a t e s o f s p i n s

|j - J | < J < j + J s p c o r e s p c o r e

We now a d d o n e l e v e l o f c o m p l e x i t y t o t h i s p r o b l e m b y c o n s i d e r i n g

h a r m o n i c o s c i l l a t i o n s o f a s p h e r i c a l c o r e . T h e Y „ s h o w n i n H . .r £ m - c o r e m t

i s a s u r f a c e s h a p e o p e r a t o r o f t h e c o r e . F u r t h e r , t h e o s c i l l a t i o n s t o

w h i c h we r e f e r a r e s h a p e o s c i l l a t i o n s . I t t h e n f o l l o w s t h a t b y t h e

s t a n d a r d c a n o n i c a l p r e s c r i p t i o n ( E i 7 5 ) we c a n w r i t e Y „ a s t h e sumr 2 m - c o r e

o f q u a d r u p o l e p h o n o n c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s .

I f we a p p l y t h i s v i a f i r s t - o r d e r p e r t u r b a t i o n t h e o r y , we f i n d t h a t

t h e m u l t i p l e t s r e m a i n d e g e n e r a t e : s i n c e o n l y o f f - d i a g o n a l m a t r i x e l e

m e n t s o f H . c a n b e n o n - z e r o u n d e r t h e f o r m o f Y „ i n t h e h a r m o n i ci n t 2 m - c o r e

o s c i l l a t o r l i m i t , e n e r g y s h i f t s i n t h e f i r s t o r d e r c a l c u l a t i o n a r e z e r o .

H o w e v e r , s e c o n d o r d e r p e r t u r b a t i o n t h e o r y c a n g i v e e n e r g y s h i f t s f r o m

t h e m i x i n g o f s t a t e s u n d e r H. S i n c e H . . m i x e s c o r e s t a t e s d i f f e r i n gy m t i n t

i n p h o n o n n u m b e r b y o n e , t h i s s e c o n d o r d e r e f f e c t c a n b r e a k t h e d e g e n e r

a c y o f t h e m u l t i p l e t s . T h e d e t a i l s o f t h e h a r m o n i c o s c i l l a t o r t r e a t m e n t

c a n b e f o u n d i n ( E i 7 5 ) .

F o r t r a n s i t i o n a l n u c l e i w h o s e c o r e s a r e o n l y s l i g h t l y d e f o r m e d t h e

n o n - z e r o m a t r i x e l e m e n t s Y „ b r e a k t h e d e g e n e r a c i e s i n f i r s t o r d e r .2 m - c o r e

I n t h i s c a s e , t h e f i r s t o r d e r e n e r g y p e r t u r b a t i o n i s w r i t t e n (G a 8 0 b , De

6 1 )

-49-

-50-

A = (4ir/5)xf2x(-l)P x

< j | | Y < 2 > | | j > x < J | | Y ( 2 > | | J > xs p s p c o r e c o r e

J J 2 c o r e c o r e

3 s p 3 s p

(2.5.5)

w h e r e P = j s p + J . T h e s i g n s o f t h e r e d u c e d m a t r i x e l e m e n t s i n ( 2 . 5 . 5 ) a r e

c h a r a c t e r i s t i c o f t h e p h y s i c a l s i t u a t i o n . F o r a p r o l a t e c o r e s h a p e ,

< J | | Y < 2 > | | J > <0c o r e c o r e

a n d f o r a n o b l a t e c o r e s h a p e ,

< J || Y ( 2 > || J > > 0 .c o r e c o r e

A v a l e n c e p a r t i c l e c o u p l e d t o t h e c o r e g i v e s

< j || Y < 2 > || j > > 0 .s p s p

I f a h o l e i s c o u p l e d t o t h e c o r e i n s t e a d , t h e n

< j I I Y ( 2 > || j > < 0 .J s p s p

G i v e n t h e s i g n s o f t h e s e r e d u c e d m a t r i x e l e m e n t s , t h e p h a s e a n d s i x - j

s y m b o l d e t e r m i n e t h e o r d e r i n g o f t h e s t a t e s i n t h e m u l t i p l e t . A n e x a m

p l e o f s u c h a m u l t i p l e t w i t h b r o k e n d e g e n e r a c y c a n b e f o u n d i n f i g u r e

2.11.T h e s i n g l e p a r t i c l e r e d u c e d m a t r i x e l e m e n t o f e q u a t i o n ( 2 . 5 . 5 ) c a n

b e c a l c u l a t e d f r o m a s i m p l e e x p r e s s i o n t h a t m ay b e f o u n d , f o r e x a m p l e ,

i n ( B r 7 7 ) . T h e v a l u e s f o r t h e c o r e m a t r i x e l e m e n t c a n e i t h e r b e t a k e n

f r o m d a t a o n t h e c o r e o r c a l c u l a t e d . One w a y o f c a l c u l a t i n g t h i s (G a

F i g u r e 2 . 1 1 W e a k c o u p l i n g m u l t i -

p l e t s f o r j s p = 9 / 2 h o l e a n d p a r t i c l e

c o n f i g u r a t i o n s .

WEAK COUPLING MULTIPLETS

3 /2 H

5/2‘

9 /2

W/Z

13/2

9 /2

9g/2 hols

■ l/2+— h17/2+

. 7/2+--------------

,l5/2+*

l3 /2+*

4 +

g9/2 particle

----------------l3/2+

15/2“

7/2+

l/2+9/2+

l7/2+5/2+

/2+3 /2 +

h 5/2+■

f 7/2+*

Il/2+-

2+11/2 A

9/27/2 13/2 5/2+

9 /2 + 9 /2 +

8 0 b ) u s e s a g e n e r a l i z e d s e n i o r i t y s c h e m e , w h i c h i s a m e t h o d f o r g e n e r a t

i n g c o l l e c t i v e b e h a v i o r w i t h i n a s h e l l m o d e l f r a m e w o r k ; t h i s p a r t i c u l a r

a p p r o a c h h a s b e e n u s e d s u c c e s s f u l l y t o a c c o u n t f o r t h e b e h a v i o r o f o d d - A

I o d i n e i s o t o p e s (G a 8 2 ) .

We u s e t h e t e r m i n t e r m e d i a t e c o u p l i n g t o d e n o t e a s i t u a t i o n i n

w h i c h s e c o n d - o r d e r p e r t u r b a t i o n e f f e c t s f r o m t h e h a r m o n i c o s c i l l a t o r

p a r t o f t h e p a r t i c l e - c o r e i n t e r a c t i o n a s w e l l a s t h e e x c h a n g e i n t e r a c

t i o n , t h e e f f e c t t h a t t h e v a l e n c e n u c l e o n h a s o n t h e c o r e v i a t h e P a u l i

P r i n c i p l e , b e c o m e i m p o r t a n t . T h e m u l t i p l e t s t r u c t u r e b e g i n s t o b r e a k

d o w n , a n d s t a t e s m i x g e n e r o u s l y w i t h s t a t e s h a v i n g d i f f e r e n t p h o n o n num

b e r s ( i . e . a r i s i n g f r o m d i f f e r e n t c o r e s t a t e s ) . One p a r t i c u l a r l y

s t r i k i n g s i g n a t u r e o f i n t e r m e d i a t e c o u p l i n g i s t h e o c c u r e n c e o f t h e " j - 2

a n o m a l y " , t h e p h e n o m e n o n o f a b a n d h e a d h a v i n g a s p i n l e s s t h a n t h a t o f

t h e s i n g l e p a r t i c l e o r b i t a l t o w h i c h i t c o r r e s p o n d s i n a n u c l e u s t h a t

o t h e r w i s e a p p e a r s w e l l d e s c r i b e d b y w e a k c o u p l i n g .

2 . 6 S T R O N G C O U P L I N G I N O D D - A N U C L E I A N D T H E O C T U P O L E

N I L S S O N M O D E L

S t r i c t l y s p e a k i n g , t h e s i n g l e p a r t i c l e a n g u l a r m omentum j i s a g o o d

q u a n t u m n u m b e r i n t h e i n t r i n s i c s y s t e m o n l y w hen n o s t a t i c d e f o r m a t i o n

e x i s t s i n t h e c o r e . H o w e v e r , a s t h e t r e a t m e n t a b o v e i n d i c a t e s , r e g a r d

i n g j a s a g o o d q u a n t u m n u m b e r i n t h e i n t r i n s i c f r a m e i s s t i l l u s e f u l a s

l o n g a s t h e s t a t i c d e f o r m a t i o n i s s m a l l ; i n o t h e r w o r d s , t h e w e a k

c o u p l i n g b a s i s , w h i c h i s b u i l t a r o u n d a s p h e r i c a l n u c l e u s , i s u s e f u l a s

-52-

H o w e v e r , f o r a w e l l - d e f o r m e d n u c l e u s t h e p e r t u r b a t i o n a p p r o a c h

b r e a k s down c o m p l e t e l y a n d N i l s s o n ' s m e t h o d f o r c a l c u l a t i n g t h e d e f o r m e d

s h e l l m o d e l ( N i 5 5 ) m u s t b e u s e d i n s t e a d . T h e N i l s s o n m o d e l i s b a s e d

p r i m a r i l y o n tw o p r e m i s e s . F i r s t , t h e g o o d q u a n t u m n u m b e r p e r t i n e n t t o

t h e t r e a t m e n t o f s i n g l e p a r t i c l e o r b i t a l s i n a d e f o r m e d s y s t e m i s Q, ,

t h e p r o j e c t i o n o f t h e s i n g l e p a r t i c l e o r b i t a l a n g u l a r m omentum o n t h e

s y m m e t r y a x i s o f t h e d e f o r m e d n u c l e u s . S e c o n d , t h e i n t e r a c t i o n o f t h e

s i n g l e p a r t i c l e o r b i t a l w i t h t h e c o r e d e p e n d s c r i t i c a l l y o n t h i s o r i e n

t a t i o n . F o r e x a m p l e , t h e q u a d r u p o l e m om ent o f a K = l / 2 s i n g l e p a r t i c l e

o r b i t a l i s m o re n e a r l y a l i g n e d w i t h t h e q u a d r u p o l e m om ent o f t h e c o r e

t h a n i s t h a t o f a n o r b i t a l w i t h a l a r g e r K . T h e o u t p u t o f a N i l s s o n

m o d e l c a l c u l a t i o n i s a N i l s s o n d i a g r a m , a p l o t o f t h e e n e r g i e s o f s i n g l e

p a r t i c l e o r b i t s a s a f u n c t i o n o f a d e f o r m a t i o n p a r a m e t e r d e s c r i b i n g t h e

c o r e a s w e l l a s t h e c o r r e s p o n d i n g s i n g l e p a r t i c l e w a v e f u n c t i o n s .

N a z a r e w i c z a n d O l a n d e r s h a v e d e v e l o p e d a N i l s s o n - l i k e m o d e l t h a t

u s e s t h e W o o d s - S a x o n p o t e n t i a l ( i n s t e a d o f t h e m o d i f i e d h a r m o n i c o s c i l

l a t o r u s e d f o r t h e N i l s s o n m o d e l ) a n d t a k e s a s t a t i c o c t u p o l e s h a p e i n t o

a c c o u n t t o g e t h e r w i t h b o t h s t a t i c q u a d r u p o l e a n d h e x a d e c a p o l e d e f o r m a

t i o n s (N a 8 5 ) . I n t h i s c a l c u l a t i o n , t h e y f i x e d 3 2 a n d t o v a l u e s

w h i c h t h e y t a k e t o r e p r e s e n t a v e r a g e s o v e r t h e m a s s r a n g e A = 2 1 9 - 2 2 9 . B y

f i x i n g p 2= 0 . 1 5 a n d 34 = 0 . 0 8 , t h e y h a v e r e p r o d u c e d r e c e n t l y m e a s u r e d

g r o u n d s t a t e s p i n s ( A h 8 3 ) i n o d d - A R a i s o t o p e s w i t h A = 2 21 - 2 2 9 . T h e

e n e r g i e s o f a n u m b e r o f o r b i t a l s i m p o r t a n t i n R a i s o t o p e s a r e p l o t t e d

a g a i n s t (5 i n f i g u r e 2 . 1 2 . T h i s f i g u r e a l s o i l l u s t r a t e s t h e t h e o r e t i

c a l l y p r e d i c t e d g r o u n d s t a t e s p i n s f o r 2 1 9 - 2 2 9 R a b y u s i n g v a l u e s o f

-53-

long as perturbation theory is appropriate.

Figure 2.12 The Woods-Saxon single particle neutron (a) and proton (b) orbitals plotted versus octupole deformation The other defomation

parameters, fl =0.15 and cor

respond approximately to calculated (see figure 2 .8) equilibrium deformations for the transitional Ra-Th nuclei. The arrows indicate the predictions for measured (221_229Ra,2 2 3 F r / 2 2 5 - 2 2 7 A c a n d 2 2 7 - 2 2 9 p a ) a n d

unmeasured (219Ra) ground state spins.

I j ft A = 222-------------- 1----- 1----- 1----- 1----- 1----- 1----- 1----- 1----- 1----- 1----- j

-54-

Such a deformed single particle model can, in principle, be con

structed for a cluster configuration as well. However, this work has

not yet been done.

2.7 P A R I T Y D O U B L E T S IN R E F L E C T IO N A S Y M M E T R I C O D D - A N U C L E I

A static intrinsic reflection asymmetric shape in any form gives

rise to a very striking phenomenon in an odd-A nucleus, the parity doub

let (Ch 80). When the shape of a nucleus is reflection symmetric, par

ity is a good quantum number in the intrinsic frame. However, this is

not true for a reflection asymmetric case. The result in the laboratory

frame is a doublet of states with identical intrinsic structure and spin

but opposite parity. The degeneracy of this doublet is broken by the

same mechanism that displaces the positive and negative parity bands in

an even-even static reflection asymmetric nucleus, the finite size of

the potential barrier separating the two octupole deformed potential

energy minima (Le 82a)(see figure 2.9).

Such parity doublets have been observed in a number of odd-A nuclei

near A=225; an example is displayed in figure 2.13. At this time, these

parity doublets are the most substantial experimental evidence that

reflection asymmetric shapes exist in a static (instead of vibrational)

form.

-55-

predicted in (Na 85).

Figure 2.13 Level spectrum of 225Ac taken from (Ah 84) . Lines connect parity doublets. No parity doublet partner was observed for the 9/2+ state at 144.90 keV.

-56-

Reference :

9 / 2 -

7 / 2 -

5 / 2 -

7 / 2 -

5 / 2 -

3/2-

I. Ahmad, et al Phys . Rev. Lett. 5 2 , 5 0 3 ( 1 9 8 4 )

2 3 5 . 5

9 / 2 + 2 5 6 . 9

7 / 2 +

1 7 0 .8

1 2 0 .8 07 / 2 +

1 9 9 .9 0

1 5 5 . 6 51 4 4 . 9 0

1 0 5 . 5 0

77 .15 / 2 +

895A c136

6 4 . 7 0

40.09

3. EXPERIMENTAL PROCEDURE

3.1 G E N E R A L F E A T U R E S

All three of the nuclei studied herein were produced by fusion-eva-

poration reactions in the 14C+208Pb system; 216Rn, 218Ra and 220Ra are

products of the 2 0 8Pb(14C,ct2n), 208Pb(14C ,3n) and 208Pb(14C ,2n) chan

nels, respectively. We measured the gamma-ray excitation function at

beam energies of 60 MeV through 78 MeV, gamma-gamma coincidences at 68

MeV, gamma-X-ray coincidences at 68 MeV and gamma-ray angular distribu

tions at 67 MeV. Both thin (300 microgram/cm2 208Pb on 225 microgram/

cm2 Au) and thick (50 milligram/cm2) targets were used for the excita

tion function measurement. All other measurements were performed with

the thick target only.

The 14C ions were prepared in a cesium sputter ion source (Mi 74)

and accelerated in the Brookhaven National Laboratory MP-7 Tandem Van de

Graaff accelerator. The accelerated beam was momentum analyzed in a 90

degree magnet, directed to the experimental area by a subsequent switch

ing magnet and adjusted on target with by a series of quadrupole focus

ing magnets and electrostatic beam steerers.

Glass test tubes with 0-ring stoppers and plunger-like target hold

ers were used as target chambers. For the thin target excitation curve

measurement ports in the test tube allowed for the entrance and exit of

the beam. The beam was then stopped several meters downstream and the

current collected in an insulated Faraday cup connected to a beam cur-

For the thick target experiments, only an entrance port was needed

in the target chamber since this target stopped the beam. The beam cur

rent was collected directly from the target holder in this configura

tion .

A variety of Ge(Li) and high purity n-type Ge gamma-ray detectors

were used. The energy calibrations and relative efficiencies of the

detectors were determined with 60Co, 133Ba and 152Eu sources.

The thin target was produced by resistively heating 99.99% isotopi-

cally pure 20 8Pb in a Ta boat and evaporating, under vacuum, onto a Au

foil mounted on a target frame. Target thickness was determined with

the use of a deposit thickness monitor (DTM). Such a measurement is

generally reliable to within 40%. The thick target was mechanically

rolled.

Signals taken from detector preamplifiers were processed using

standard NIM electronics. Processed pulses carrying timing and energy

information were passed to analog-to-digital converters and subsequently

read by the Brookhaven Tandem Laboratory Xerox Sigma-7 computer. Coin

cidence data were stored on tape in event-by-event form for further

off-line sorting; direct spectra were stored in 4096 channel form.

Off-line coincidence sorting and peak-fitting were performed using the

Wright Nuclear Structure Laboratory IBM 4341 computer as well as the

Brookhaven Sigma-7.

-58-

rent integrator (BCI).

-59-

We should now discuss briefly the properties of heavy ion fusion-e-

vaporation reactions. The reaction can be described in a simple way as

follows. The projectile, in our case 14C, approaches the target, 208Pb.

If the center of mass energy of the system, Ecm, is such as to exceed

the electrostatic repulsion, then there is a high probability that the

two nuclei will fuse to form a compound nucleus, 222Ra. For values of

Ecm required for this in the system under study the resultant compound

nucleus is left with an excitation energy of several times the neutron

separation energy. Consequently, the nucleus evaporates several parti

cles (mostly neutrons) and emits a number of gamma rays. After the

excitation energy is reduced to the neutron separation energy, the

nucleus emits gamma rays until the ground state is reached.

Phenomenologically, the cross section for a particular evaporation

channel, r, can be written as a product,

o (E ) = I (E )o„(E ) (3.2.1)rx cm r cm T cm' '

where oM E ) is the total fusion cross section, which depends largely T cm

on electrostatic and angular momentum considerations, and Ir(E ) repre

sents the probability that the fused nucleus will choose a particular

decay channel. The function °x(Ecm) rises sharply at the energy

required to overcome the mutual electrostatic repulsion, called the Cou

lomb barrier, which can be estimated to be %

3 .2 HEAVY ION FUSION-EVAPORATION REACTIONS

ECB = Z1Z2 1 ( Al 1/3+ A21/3) (MeV) <3 -2 -2>

-60-

The maximum of the function I (E ) can be estimated to be locatedr cm

at

E = -Q + I. a, (3.2.3)max-r r k k '

where Q is the Q-value for channel r and a, is the kinetic energy of IT k

the kth particle evaporated. Neutrons are evaporated with an average

kinetic energy of 6 MeV. Corresponding values for charged particles are

a few MeV higher.

In order for a channel r to have an appreciable cross section, two

conditions must be fulfilled. First, I (E ) must be relativelyr max-r

large. For compound systems that are less than approximately ten neu

trons from the line of maximum stability the Coulomb interactions among

charged particles in the compound system act to enhance the probability

for the evaporation of a neutron relative to that for a charged particle

(Ne 74). As we note in chapter 4, we observe that the probability for

the evaporation of a neutron in the system under study is roughly 70

times that for the evaporation of an alpha particle. The proton prob

ability is somewhat lower. Therefore, the evaporation of several neu

trons and no charged particles characterizes the dominant channels.

Second, E must be located near or above E__. An example of themax-r Co

behavior of neutron evaporation cross sections (tracing what are refer

red to as Ghoshal curves) is given in figure 3.1.

The most successful calculations of such fusion-evaporation cross

sections are performed using a statistical model (Ga 80a). The computer

code PACE utilizes this method and is available on the WNSL IBM 4341.

As in figure 3.1 , the 2n channel cross section for the 14C+208Pb

Figure 3.1 Excitation function for 181Ta(l60,xn) reaction (Ne 74).

80 100 120 140 160 180 200LAB ENERGY (MeV)

Fig. 6. Excitation functions for the 181Ta(ieO, xn)197_a:Tl reactions calculated with the program of Sikkeland (1967) (redrawn with permission from Newton, “Progress in Nuclear Physics,” 1969, Pergamon Press Ltd.).

system is critically dependent on the relationship between E andC B

E . With compiled nuclear masses we find that max-r

Q = -45 MeV, xr

which results in

-62-

E = 5 7 MeV.max-r

The Coulomb barrier is

Eori = 59 MeV.CB

Using PACE, we predict that the 2n cross section reaches a maximum of 20

mbarn, a reasonable cross section for spectroscopy.

The Q-value is as large as it is (in absolute value) because both

the target and projectile in this system are so tightly bound. If we

replace the target with another that is less tightly bound, the calcu

lated 2n cross section plummets. As an example, the 14C+210Pb system

has

E = 5 4 MeVmax-r

and

E„„ = 59 MeV.CB

The 2n yield curve falls below the Coulomb barrier and the maximum cross

section, as predicted with PACE, is less than 1 mbarn.

Heavy ion fusion characteristically results in a compound nucleus

with large angular momentum. In our case, the compound nucleus, 222Ra,

has an average angular momentum of about 40R before decay begins.

Evaporated neutrons carry off only 1R to 2E of angular momentum each, so

the residual nucleus always possesses a high spin. Subsequent gamma ray

emission cannot carry off much angular momentum either until the nucleus

is in a state close to the yrast line, the locus of states of maximum

spin for a given energy. Once the gamma ray deexcitation cascade

reaches the yrast line, the nucleus subsequently deexcites predominantly

via stretched dipole and quadrupole transitions until the ground state

is reached. Because of this, yrast states are preferentially populated

and non-yrast low spin states are very weakly populated in heavy ion

induced reactions.

Since the incoming angular momentum varies with beam energy, one

would expect that the strongest population of low spin non-yrast states

would occur at energies close to the Coulomb barrier. This was demon

strated in a study of the 13C+208Pb system (En 84) at energies between

59 MeV and 72 MeV.

3.3 E X P E R I M E N T A L D E T A I L S

In this section, we detail targets, electronics and detector con

figurations used for each of the measurements. The thin target excita

tion function experiment was performed with a single high purity n-type

Ge detector (HPGe) having 20% efficiency (relative to a standard 3"x3"

Nal(Tl) detector) and energy resolution of 2.0 keV full width at half

maximum (FWHM) at 1.33 MeV. The detector was placed at 90 degrees rela

-63-

tive to the beam. Measurements were made at beam energies of 60 MeV, 62

MeV, 64 MeV, 66 MeV, 68 MeV, 73 MeV and 78 MeV. Relative normalization

was taken from BCI readings.

The Brookhaven National Laboratory Compton Supression Spectrometer

(CSS), composed of a Ge(Li) detector placed in the central cavity of a

25.4 cm x 20.3 cm Nal(Tl) crystal was used to measure the thick target

excitation function. The Ge(Li) detector had 2.1 keV resolution FWHM at

1.33 MeV and was 19% efficient. Again, relative normalization was taken

from BCI readings. The electronic circuit used is shown in figure 3.2.

For the gamma-gamma coincidence experiment, we used two Ge(Li)

detectors and one HPGe detector. One Ge(Li) unit had 19% efficiency and

2.1 keV resolution at 1.33 MeV; the other was 22% efficient and resolved

2.0 keV at 1.33 MeV. The HPGe had 20% efficiency and 2.0 keV resolution

at 1.33 MeV. The thick target and a beam energy of 68 MeV, where the

yield of the 2n and 3n channels reached a maximum (see figure 3.1), were

used. The electronics were arranged so that the simultaneous firing of

any two detectors was recorded as an event. Thus, we effectively

recorded data simultaneously from three pairs of detectors. The timing

circuit is shown in figure 3.3, and the energy circuit in figure 3.4. A

total of 1.7x10® coincidence events were recorded.

In order to further enhance our statistical accuracy, we extracted

data from each pair of detectors in two ways. If the two detectors in a

pair are called G1 and G2, then the spectrum of gamma rays in coinci

dence with a gamma ray A was studied by first obtaining the spectrum of

G2 gated by the detection of A in Gl, and then obtaining the spectrum of

G1 gated by the detection of A in G2. The two spectra were then added

-64-

Figure 3.2 Electronic instrumentation used for compton suppression spectrometer. Modules are standard NIM units. G+D = gate and delay generator, SCA = single channel analyzer.

Ge (Li)CENTRALDETECTOR

Nal (T6) COMPTON

SUPPRESSOR

A L IN EA R

Ge(Li) AND Nal BOTH FIRE

EVENTCOMPTON

COINCIDENCE MODULE

G+D

G+DEVENT

VETOED EVENT

ACCEPTED EVENT

ONLY Ge(Li) F IRES

IOnLn1

Figure 3.3 Electronic instrumentation used for timing signals in gamma-gamma coincidence experiment. CFD = constant fraction discriminator, TFA = timing filter amplifier, TAC = time-to-pulse height converter, LGS = linear gate stretcher.

63ns 63ns

Figure 3.4 Electronic instruments tion used for energy signals in gam ma-gamma coincidence experiment.

XIHPGe

r 2Ge (L i )

X 3 Ge (Li)

G +DE V E N T

G + DEV EN T

G + DEV EN T

IONI

to one another, and finally added to the corresponding spectra from the

other two pairs. Contributions to the gated spectrum from coincidences

with the Compton scattered background were removed by subtracting a

spectrum gated on background gamma rays at an energy near that of the

transition of interest.

The use of the thick target requires comment. Obviously, a thick

target enhances reaction yield, which is especially helpful in the cases

of the weak 2n and ct2n reaction channels. However, since the thick tar

get reduces the energy of, and then stops the beam, we simultaneously

observe reactions taking place at the beam energy of 68 HeV, at which

high spin and yrast states are populated almost exclusively, as well as

at all intermediate energies down to those near the Coulomb barrier of

59- MeV, where non-yrast low-spin states are populated significantly.

Two detectors were used for the angular distribution measurements.

The first was a planar Ge detector with 700 eV resolution (FWHM) at 122

keV. The second was the CSS with a HPGe detector substituted for the

Ge(Li) in the central cavity of the Nal(Tl). This HPGe had an effi

ciency of 33% and a FWHM resolution of 2.1 keV at 1.33 MeV. The planar

Ge had superior resolution and a very small Compton scattering yield;

however, its detection efficiency for higher energy gamma rays made it

impractical for detecting photons with energies greater than 400 keV;

the CSS was used for these. Relative normalization was performed with

the BCI. Measurements were taken with each detector at settings of 0,

15, 30, 60, 75 and 90 degrees with respect to the beam.

After correcting for detector efficiency and BCI normalization, we

-68-

fitted the angular distribution of each gamma ray to

-69-

W(e) = Ao{1+A2P2 (cos0)+A4P4 ( cos0)} (3 .3 .1 )

The expansion was terminated at the P4 term because data from only six

angles were taken; further terms in the expansion would have signifi

cantly widened the uncertainty in the determination of the fitted coef

ficients and reduced the usefulness of the results.

Energy excitation spectra were constructed and spin assignments

made using the coincidence and angular distribution information. Intra

band transitions were assumed to be stretched, a common assumption in

heavy-ion fusion-evaporation reactions (Wa 85). Stretched dipole and

quadrupole transitions are characterized by values of A2 which are neg

ative and positive, respectively.

Electron conversion coefficients provided a convenient method for

making parity assignments. Electron conversion is the process by which

a transition is internally converted so that an atomic electron emerges

rather than the photon. The relationship between the two processes can

be specified by

Ie = aly (3.3.2)

where I is the number of transitions by gamma ray emission, I is the 0 ®number of transitions by electron conversion and a is,by definition, the

electron conversion coefficient. These coefficients are tabulated in

(Ha 68) and (Dr 71) and differ substantially for different multipoles

and elements.

For gamma rays in Rn and Ra, these large conversion differences can

be exploited by recognizing that the sum of the intensities of tran

sitions feeding a state must be no more than the sum of the intensities

of transitions deexciting the same state. We will give specific exam

ples in chapter 4.

The gamma-X-ray experiment, used for identification of gamma rays

in 216Rn, was performed with a HPGe with 20% efficiency and 2.1 keV res

olution at 1.33 MeV, and a planar Ge with 700 eV resolution at 122 keV.

The HPGe was used for gating.

-70-

4 ANALYSIS AND RESULTS

4.1 P R E V I O U S W O R K ON 2 U Rn, 2 2 0 Ra A N D 219Ra

An early alpha particle decay study of the 228U decay chain (Ru 61)

provided information on both 216Rn and 220Ra. Three levels were found

in 220Ra: a firmly assigned 2+ level at 177±2 keV, a tentatively

assigned 1" level at 410±3 keV, and a tentatively assigned 4+ level at

474±3 keV. In 216Rn only one level was identified, this at 465±4 keV; a

tentative assignment of 2+ was reported.

A previous study of high spin states in 220Ra was reported in (Bu

84). The weak 208Pb(18O,a2n)220Ra reaction was used in this study, and

levels with tentatively assigned spins up to 12fi were observed.

A study of 219Ra via the 208Pb(14C,3n)219Ra reaction was published

in 1984 (Mi 84). The yrast sequence up to a spin 18h beyond the ground

state value was mapped in this study.

4.2 E X C I T A T I O N F U N C T I O N M E A S U R E M E N T S

The thin target excitation curve measurement showed quite clearly

the characteristic Ghoshal pattern of (HI,xn) cross sections for the

strong 3n and 4n channels (see figure 4.1). In fact, the cross section

maxima matched quite well those calculated with the use of the PACE

code. Gamma rays from reactions of the beam with the gold target back-

Figure 4.1 Excitation function measurements for thin and thick targets (top) and calculation using the PACE code (bottom).

Cross Section (mb) Normalized Yield (arbitrary units)

ing obscured those expected from 220Ra. However, the self-supporting

thick target gave a significant yield from 220Ra,- at 66 MeV, this yield

was roughly 10% that from 219Ra.

Not enough counts were collected in either excitation function

measurement to determine the yield of 216Rn. Instead, we deduced from a

singles spectrum - collected at 68 MeV with the thick target - that the

yield of 216Rn is 1.5% that of 219Ra under these conditions.

4.3 220Ra L E V E L S P E C T R U M

The production of 220Ra in the 14C+208Pb reaction was indicated by

the observation of the 178 keV gamma-ray identified as the 2+-»0+ tran

sition in (Ru 61). However, a 176 keV transition also deexcites the 192

keV level in 219Fr (Le 78), which would be the product of the (p2n) eva

poration channel. In order to rule out the possibility of mistaken

identification, we searched for the 192 keV deexcitation of the same

state, which is favored with a branching ratio of five to one over the

176 keV one. Such a strong 192 keV gamma- was not seen. A further con

firmation of the identity of 220Ra came from the study of the same

nucleus by the 208Pb(180,a2n)220Ra reaction (Bu 84).

From coincidence spectra and angular distribution information we

constructed the level spectrum shown in figure 4.2. The coincidence

spectrum gated on the 178 keV gamma-ray is shown in figure 4.3, and a

planar Ge singles spectrum from the angular distribution experiment is

displayed in figure 4.4. The ordinate scale in figure 4.4 is chosen to

-73-

Figure 4.2 Level spectrum of 220Ra deduced in this work.

(2ij 3623.0 1479.7

(J9J|3I4_3.32960.4 1455.4

1 (9)687.9

|428.1 (6 )

2259.8398.2 (13)

13“ *1861.6367.2 ( I I )II" *1494.4332.4 (20)1162.0290.3■(,4)87l.7238.2

5- (3)633.5(3”) 473.8

231.2(100) 2 +

178.1(105o+-

412.5295.7 234.4

2 2 0 Ro

Figure 4.3 Energy spectra from our gamma-gamma coincidence experiment. Insets in the upper right show weak transitions between low spin states observed in the 178 keV gate. Inset at the lower right displays ■ weak transitions between high spin states observed in the summed spectrum.

COUN

TS PER

CH

ANNE

L- 7 5 -

68 MeV 208Pb( l4C,2n)220Ra

O 40 360 440

CHANNEL

Figure 4.4 Singles spectrumdetected in a planar Ge detector (LEPS) as found in the angular distribution experiment. The scale for the number of counts is chosen to show transitions in 220Ra • more clearly. Many peaks from 219Ra are thus cut off by the upper boundary of this figure.

COU

NT

S

CHANNEL

IO 'I

For spins of 7h, and above, we were able to exploit the alternating

parity structure to make independent confirmation of multipolarities.

Each of these states deexcites by both a dipole and a quadrupole tran

sition. Thus, the energy of each quadrupole transition must be the sum

of the energies of the two corresponding dipole transitions. Further

more, the quadrupole transition must not be in coincidence with either

of these dipole transitions unless a doublet is involved. This informa

tion was particularly helpful for those gamma-rays that were obscured by

stronger lines from 218Ra. For these cases, as well as for gamma-rays

which are doublets with lines from 218Ra or other transitions from

220Ra, intensities were estimated from coincidence spectra. In all

other cases intensities were deduced from angular distribution informa

tion.

In chapter 3 we described the inductive argument by which we deter

mined the electric or magnetic character of transitions. Here we give

as an example the feeding and deexcitation of the 6+ state at 687 keV.

From the determination in (Ru 61) that the 178.1 keV transition is E2,

from angular distribution data and from intensity balances we can show

that both the 231.2 keV (4+-»2+) and 277.8 keV (6+-»4+) transitions are

E2's. The 6+ state is fed by the 184.5 keV L=1 and 312.9 keV L=2 tran

sitions that have gamma-ray intensities of 55±1 and 50±2, respectively,

relative to the value of I =100 assigned to the 231.2 keV gamma-ray. If0the 312.9 keV line is magnetic, then it has a total electron conversion

coefficient of approximately 2.25 and a total transition intensity I

of 111±5 (normalized so that 1^=100 for the 231.2 keV transition).

-77 -

highlight transitions in 220Ra.

Since the 277.8 keV transition, which deexcites the 687 keV state, has

only Itr=75±6, the 312.9 keV gamma-ray must be E2. Further, if the

184.5 keV line is Ml it would have a conversion coefficient of approxi

mately 2.9, thus violating intensity balance once again. On the other

hand, the assumption of electric multipolarities for both feeding tran

sitions gives conversion coefficients of 0.17 and 0.12 for the 312.9 keV

and 184.5 keV gamma-rays, respectively. The resulting intensities are

I. =40±2 for the 312.9 keV line and I. =42±1 for the 184.5 keV line, tr tr

The sum of these intensities 82±4, agrees well with the deexciting

intensitiy of 75±6.

We have altered the assignment of the 473.8 keV level from that

given by (Ru 61), which had been, tentatively, 4+ . Because the 231.2

keV E2 gamma-ray is the strongest in the 178.1 keV coincidence spectrum,

the 409.3 keV level is clearly 4+ . However, we observe in the 178 keV

gate a weak gamma-ray at 295.7 keV (see inset in top of figure 4.3) that

probably corresponds to the (4+)-»2+ transition reported in (Ru 61). The

3" assignment that we tentatively report for the 473.8 keV level is con

sistent with our data, results from (Ru 61) and the systematic behavior

in neighboring nuclei (Ga 83b, Ku 76).

In addition, there is evidence for a weak 234.4 keV gamma-ray in

the 178 keV gate. This is most probably the l'-»2+ transition observed

in (Ru 61).

In order to detect weak gamma-rays from 220Ra we added spectra

gated on six 220Ra gamma-rays: 153 keV, 162 keV, 178 keV, 185 keV, 231

keV and 367 keV (see figure 4.3). From the resulting totalled spectrum

we extracted four transitions that we tentatively placed in the 220Ra

-78-

Our results agree well with those of two other high spin studies of

220Ra (Bu 84, Ce 85). We include additional levels at 412 keV and 474

keV as well as tentative levels at 3143 keV and 3623 keV. A listing of

the properties of gamma-rays in 220Ra is given in table 4.1.

Each level above the 4+ state in our spectrum of 220Ra decays by

both El and E2 transitions. From each gamma-ray branching ratio we can

extract the ratio of the reduced matrix elements of the transitions,

B(E1: J-»J-1)/B(E2 : J-»J-2). Expressions in (Bo 69) yield the following:

B(E1 :J-»J-l)/B(E2:J-»J-2) =

{ Iy(El) / Iy (E2) } x { EE2 5/EE13 } x ( 7.71 x 10 ' 7 fm ' 2 )

where E and E_„ are the energies of the El and E2 transitions, respec- tl EZ

tively, expressed in MeV. Values for 220Ra can be found in table 4.2.

The systematic behavior of this ratio will be discussed in section 5.

4.4 216Rn L E V E L S P E C T R U M

The tentative assignment of a 465±4 keV gamma-ray as the 2++0+

transition of 216Rn in (Ru 61) prompted us to carry a systematic inves

tigation of gamma-rays between 460 keV and 470 keV in an effort to iden

tify the corresponding transition in our spectrum. We focused on a

461.9 keV gamma-ray that had not been assigned to any of the stronger

channels. When we examined the spectrum from our X-ray-gamma experiment

gated on this gamma-ray, both the 81.1 keV and 83.8 keV Rn K X-rays were

-79-

level spectrum. They are denoted by dotted lines.

-80 -

Table 4.1 CHARACTERISTICS OF 22 0 Ra GAMMA RADIATION

Gamma ray energies (E ), angular distribution coefficients from the0expression

W(0) = Ao {1+A2P2 (cos0)+A4P4 ( cos0)}

gamma ray intensities (1^)> total transition intensities (Itr) an^

transition assignments (jT-J 1 ).

E (keV) a) A2 A4 I b) Ifcr c) JV j1f

128.2 -0.23( 3) 29(1) 27(1) 8+ - 7“

151.8 —0.16( 2) 22(1 ) 19(1) 13“ - 12 +

153.2 -0 .22( 2) 30(1) 25(1) il - 10 +

156.5 -0.47( 5) 10(1 ) ■ 8(1) ls- - 14+

162.1 -0 .22( 1 ) 43(1) 35(1) 9“ - 8+

166.6 e) 5(1) 4(1) 17“ - 16+

178.1 d) 73(4) 105(5) 2+ - 0+

179.0 d) 34(4) 26(4) 10+ - 9"

184.5 -0 . 2 1( 1 ) 55(1) 42(1) 7“ - 6+

215.4 -0.24( 4) 23(1) 17(1) 12+ - 1 1"

224.2 0.25(20) 15(1) 1 1(2) 5“ - 4+

231.2 0.29( 1) -0.07( 2) 100 100 4+ - 2+

234.4 e) 1.0(3) 0.7(2) 1- - 2+

238.2 e) 3(1) 3(1) 7“ - 5"

241.6 -0.11 (3) 23(1) 17(1) 14+ - 13-

261.2 e) 5(1) 4(1) 16+ - 15-

• -81-

271.9 -0.49(49) 16(3) 1 1(2) 18+ - 17-

277.8 e) 89(6) 75(6) 6+ - 4+

290.3 e) 17(3) 14(3) 9" - 7-

295.7 e) 2(1) 2(1) O') - 2+

312.9 -0.06( 4) 0.02( 5) 50(2) 40(2) 8+ - 6+

332.4 0.42( 2) -0.30(25) 25(2) 20(2) 1 1" - 9'

341.2 0.39(10) 0.10(13) 12(1 ) 9(1) 10+ - 8+

367.2 0.29( 9) -0.30(12) 15(1) 1 1(1 ) 13- - 1 1-

368.7 0.20(12) 0.19(18) 1 1(1) 8(1) 12+ - 10+

393.4 e) 1 2(2) 9(2) 14+ - 12+

398.2 0.36( 3) -0.29( 4) 18(1) 13(1) is - 13-

418.0 f) 4(1) 3(1) le* - 14+

428.1 e) 8(2) 6(2) 17" - 15'

439.1 <1 <1 18+ - 16+

455.4 e) 13(4) 9(3) (19-) - 17-

479.7 e) (21-) - (19‘)

a) Energies are accurate to ±0.2 keV

b) Bare y-ray intensity relative to 231.2 keV y-ray

c) Intensity (including internal conversion) relative to 231.2 keV

transition

d) Doublet in 220Ra

e) Doublet in 219Ra

f) Doublet in 218Ra

-82

Table 4.2 B(E1:J+J-l)/B(E2:J+J-2) Values in 220Ra

The spin and parity of the initial state are given by J1T.

Results are calculated from equation (4.2.1).

J11 B(E1)/B(E2) (10' 5 fm'2)

7‘ 1.72(57)

8+ 0.64(34)

9- 0.94(17)

10+ 1.76(25)

11' 1.04(09)

12+ 1 .10(1 1)

13- 2.16(18)

14+ 0.99(17)

15- 1.13(13)

16+ 1.93(63)

17- 5.19(151)

18+ >10.00

found. In-order to build up enough counts to form a recognizable peak

we compressed the energy scale of the planar Ge spectrum (see figure

4.5), sacrificing the superior energy resolution of this detector. As

shown in figure 4.5, we have reduced the effective resolution to 2 keV

per channel. However, this scale is sufficient to distinguish the 81.1

keV-83.8 keV Rn K X-ray doublet from the 83.2 keV-86.1 keV and 85.4

keV-88.5 keV K X-ray doublets of Fr and Ra, respectively. The energy

scale is also adequate to eliminate the possibility that the peak we

observe results from accidental coincidences with 84.5 keV and 84.9 keV

K X-rays from the Pb target.

The gamma-gamma coincidence 461.9 keV gate spectrum is displayed in

figure 4.6. Three strong gamma-rays are found at 378.9 keV, 385.4 keV

and 419.4 keV, respectively. A weaker line at 465.9 keV is also

observed. In addition, the 378.9 keV gamma-ray, the strongest in this

gate, can also be seen in coincidence with the Rn K X-rays despite the

weakness of the yield. A number of counts from the 205.1 keV, 234.3

keV, 294.8 keV and 414.0 keV transitions in 219Ra are also found in the

461.9 keV gate, because the tails of the 458.6 keV and 465.4 keV peaks

from 219Ra encroach on the 461.9 keV peak in the gating spectrum.

Angular distribution data indicate that the 385.4 keV and 461.9 keV

transitions are quadrupole in character. Each of the other transitions

assigned to 216Rn are obscured in singles spectra by a doublet from a

stronger channel. We can deduce that this level has natural parity

because the 461.9 keV level is populated by alpha particle decay from

22®Ra. Therefore, it has a spin-parity of 2+ and the 461.9 keV tran

sition is E2 in character. The remaining gamma-rays appear, from our

-83-

Figure 4.5 X-ray spectrum gated on the 461.9 keV transition (2+-»0+ transition in 216Rn).

COUN

TS

Figure 4.6 Gamma-ray spectrumgated on 461.9 keV transition.

COUN

TS600 500

400 300 2 0 0

1 0 0

0

10 0 200 400 600 800 1000CHANNEL

I I I r

Gate on 461.9 keVv+

2 l 6 R n : 2 +- 0

CDD-r o m

o or o

2'9Racontaminants

coincidence data, to occur within a single band with the 461.9 keV

gamma. The ordering of the gamma-rays is determined by their intensity

in the 461.9 keV gate.

The tentative spin assignments given for energy levels above the

461.9 keV state are based on the systematic behavior of neighboring nuc

lei. From the yrast spectra of these neighbors we can deduce that the

level at 840.8 keV has J1 " or 4+. Because the . 4+ state is populated

much more strongly than the 3" in heavy ion studies of the Ra and Th

isotopes, we make the assignment J1T=(4+). Identical arguments lead us

to favor the assignment of 6+ over 5" for the 1226.2 keV level and 8+

over 7" for the 1645.6 keV level.

The tentative assignment of a 10+ level at 2111.5 keV is based on

the observation of a relatively weak, but quite distinct, gamma-ray of

465.9 keV in gates on each of the lower four lines. The spin-parity

assignment again is based upon systematic behavior in this mass region.

Our final level spectrum is shown in figure 4.7; intensities shown

include electron conversion. The intensity of the 461.9 keV line is

taken directly from singles data, and for that reason, it includes

counts from the alpha particle decay of 220Ra. The yield from 220Ra was

roughly eight times that from 216Rn. Since approximately 1% of the

220Ra decays go to the 2+ level in 216Rn, roughly 8% of the intensity of

the 461.9 keV line that is shown results from the alpha decay.

Properties of gamma-rays in the 216Rn level spectrum are listed in

-86-

table 4.3.

Figure 4.7 Level spectrum of 216Rn deduced in this work.

(10 + ) - - n 21 1 1.5 -87-

( 8 + )

(6 +)

(4+)

2 +

465.9(13)

1645.6419.4(57)

1226.2385.4(60)

840.8378.9( 6 8 )

461.9

461.9( 1 0 0 )

0 + 0

2 1 6 Rn

Table 4.3 CHARACTERISTICS OF 216Rn GAMMA RADIATION

-88-

Gamma ray energies (E ), angular distribution coefficients from the0expression

W(0) = Ao{l+A2P2(cos0)+A4P4 (cos0)}

gamma ray intensities (I^)» total transition intensities (!tr) and

transition assignments (jL-J11 ).

E (keV) a) A2 A4 I b) Ifcr c) JV j1Tf

378.9 a) 67(4) 68(6) (4+) - 2+

385.4 0.27(6) -0.11(14) 59(3) 60(3) (6+) - (4+)

419.4 b) 56(3) 57(5) (8+) - (6+)

461.9 0.10(4) -0.08(9) 100(5)- 100(5) 2+ - 0 +

465.9 a) c) 13(2) (10+) - (8+)

a) Doublet in 219Ra

b) Doublet in 218Ra

c) Multipolarity unknown

d) Energies are accurate to 0.2 keV

e) Bare gamma ray intensity relative to 461.9 keV gamma ray

f) Intensity (including internal conversion) relative to 461.9 keV

transition

-89 -

We used a thin target excitation curve to make an identification of

the strongest gamma-rays from 219Ra. Since detailed information on the

4n channel (218Ra) was available (Fe 82, Ga 83b, En 84) and the 2+-»0+

transition in 220Ra (the 2n channel) was known from the alpha particle

decay work of (Ru 61), the excitation function measurement was suffi

cient to identify the strongest gamma-rays belonging to 219Ra.

Several coincidence spectra gated on gamma-rays in 219Ra are shown

in figure 4.8, and spectra from the angular distribution experiment are

displayed in figure 4.9. The ordinate scale on the planar Ge spectrum

in figure 4.9 is chosen to highlight lines in 219Ra. In figure 4.10 we

show the resulting level spectrum.

We have taken a ground state spin assignment from an examination of

the systematic behavior of ground state spins in neighboring odd-N

even-Z nuclei (see figure 4.11). For N<134, known ground state spins

are generally thought to be constant within each set of isotones (Le 78,

Ah 83). Based on the 9/2+ ground state spin assignment for 217Rn, we

tentatively assign the same spin and parity for the ground state of

219Ra.

To make spin and parity assignments for the ground state band we

made the usual assumptions for (HI,xn) reactions (Wa 85); in particular,

that intraband transitions are stretched. In light of this assumption,

angular distributions and intensity arguments allowed straightforward

spin and parity assignments for this band. A conversion electron meas

urement (Co 86) supports these assignments.

4.5 219Ra LEVEL SPECTRUM

Figure 4 .8 Gamma ray spectra gated on several t rans i t ions in 219Ra.

12

9 6

8 0

6 4

4 8

3 2

1 6

0

6

5

4

3

2I

06 0

00

4 0

8 0

20

6 0

0

-90-

~l 1 T

I 8 ™' 8 S UEs'l = 1IC U iin L . -Jl

” i— i— i— r 1 4 c + 2 0 8 p b

68 MeV '234 keV GATE l3/2+—- 9 /2 +

CDin m £i i

0 i n1

mCVJID"1 I

131 keV GATE

31/2'— 2 9 /2 +

rr ocvj

C*

OOor o

cvjOrOm0>CVJ

3 8 7 keV GATE S ID E BAND

(23/2")— (19/2”)

100 200 300 4 0 0 5 0 0

CHANNEL600 700 800 900 1000

Figure 4.9 Singles spectra from the two detectors used in the angular distribution experiment. The ordinate scale of the LEPS spectrum (top) has been chosen to show the largest peaks in the spectrum clearly.- The HPGe n-type (gamma-x) spectrum is shown at the bottom of the figure.

CO

UN

TS

xIO

3 C

OU

NT

S x

10

3 5 0

3 1 5

2 8 0

2 4 5

210 1 7 5

1 4 0

1 0 5

7 0

3 5

0 1 4 4

1 2 8

112 9 6

8 0

6 4

4 8

3 2

16

350 7 5 0 —I—

CLi nr-

\

i . J l .

C H A N N E LI 1 5 0 1 5 5 0 1 9 5 0 2 3 5 0 2 7 5 0 3 1 5 0 3 5 5 0

r* t r i

m ^ r' —? L LC\j +1 nfs £ ,1CJ ag 9 -2+ 8 ro

ro r o

v.»IS

ja. ro

cvj | jyi j "CM£ I b

,»\CM

i noCVJ

I

“i— r3950

3 cvj

m

,tN5S

„ o* o: o

O «—» t i ivN I J ftl *1 N N l' S fr* v. cm gj ♦'ki ir> .■> 'I5 c\j

(ONJ '

mro

jSirO I * 'S 'CVJ

a LEPS

T I I t 14 C + 208pb67 MeV

SPECTRUM

GAMMA-X SPECTRUM

1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 0 0 0

CHANNEL2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0

Figure 4.10 Level spectrum o f 219Ra deduced in this work.

-92-

4913.6

1 5 8 4 .7

{53/2+)-| 5-4^ -

4 9 / 2 +

(11/2“)

-TENTATIVE FROM SYST.

8 5 9 . 9

Figure 4.11 Ground state spins of even-Z odd-N nuclei in the 82<Z<92 region. Assignments shown in parentheses are tentative. Data from (Le 78, Ah 83, El 79, To 79, Ma 77b, Ma 77c, Sc 83, Ma 78).

\ nz \ 127 129 131 133 135 137 139 141

J3CLCMCO 9/2+ (9/2+)

oCL00 9/2+ 9/2+ (9/2+)

8 6 ^n(9/2+)(9/2+) 9/2+ (5/2+)(7/9/) 1 2 1

88 (9/2+) (9/2+)(9/2+) 5/2+ 3/2+ l/2+ 3/2+ 5/2+

9 0 T h (3/2+) (3/2+ ) 5/2+ 5/2t+)

9 2 u (3/2+) (5/2~) 5/2+

GROUND STATE SPINS Z > 82

The existence of the transition from the 1263.7 keV level to the

1035.6 keV level was deduced from the observation of transitions in the

ground state band above the 495.4 keV level in gates on gamma-rays in

the side band above the 1263.7 keV level. Since we did not observe a

227 keV gamma-ray in gates on the appropriate side band transitions, we

conclude that this transition has a large electron conversion coeffi

cient. For a 227 keV gamma-ray, conversion coefficients are 0.054 for

El, 0.397 for E2, 1.72 for Ml and 7.04 for M2. Further, a 227 keV M2

transition would be very slow. Consequently, the 227 keV transition

must be Ml. The determination of this multipolarity places a constraint

on the possible spins of the 1263.7 keV level.

We can also use conversion coefficients to deduce that the 128.7

keV transition must have an El multipolarity. The 414 keV gamma-ray is,

by the assumption of stretched intraband transitions and the long mean

life of an M2 transition of this energy, of E2 character. Further, we

expect from the behavior of other states in our study that the feeding

of the 128.7 keV level by the 414.0 keV transition (1^=52 relative to

the 294.8 keV transition) accounts for most of the population of this

state. The conversion coefficients for 128.7 keV gamma-rays of El, E2,

Ml and M2 multipolarities are 0.24, 3.24, 7.93 and 47.8, respectively.

Since the 128.7 keV gamma-ray has I =67 (relative to the 294.8 keV gammacTray), these multipolarities would yield, respectively, Itr=69, 235, 494

and 2702. Clearly, only an El multipolarity is consistent with our

expectation concerning the feeding of the 128.7 keV state.

Finally, since the 414.0 keV, 333.6 keV and 387.4 keV gamma-rays

appear - from angular distribution data - to be stretched E2 tran

-94-

sitions, the constraints that we have already mentioned require that the

128.7 keV level have spin-parity 11/2". This conclusion is consistent

with the observed angular distribution of the 128.7 keV gamma-ray. All

these spin assignments are listed as tentative, however, because of

their heavy dependence on intensity arguments.

Where doublets make the determination of intensities from angular

distribution data impossible, they were deduced from coincidence infor

mation. Our results are entirely consistent with the more limited

results of (Mi 84). Table 4.4 lists properties of 219Ra gamma-rays, and

table 4.5 contains corresponding B(E1: J-»J-1)/B(E2: J-»J-2) values.

-95-

Gamma ray energies (E ) , angular d i s t r ib u t io n c o e f f i c i e n t s from the

expression

-96 -

Table 4.4 CHARACTERISTICS OF 219Ra GAMMA RADIATION

w(e) = aq { i +a2p2 ( c o s 0 ) + A 4 P 4 ( c o s 0 ) }

gamma ray in t e n s i t i e s ( I . ) , to ta lA

trans i t ion in t e n s i t i e s (I . ) and trIT TTt rans i t ion assignments (J ^-J .

Ey (keV) a) A2 A4 ** b) I* c) tr 2x (J1Ii -»J

122.5 -0 .2 7 (1 ) 145(7) 155(8) 27’ - 25+

128.7 -0 .2 4 (1 ) 67(3) 69(3) (11-) - 9+

131.4 -0 .2 7 (1 ) 77(4) 79(4) (31-) - 29+

141.5 -0 .2 7 (1 ) 327(16) 324(16) 21+ - 19'

159.6 a) 433(22) 408(20) 23" - 21+

159.6 a) 72(10) 68(9) 35 ' - 33+

187.7 -0 .2 5 (2 ) 2 2 ( 1 ) 2 0 ( 1 ) 39- - 37+

196.3' 0 .23(1 ) 0 .11(1) 101(5) 90(5)

205.1 - 0 . 2 2 ( 1 ) 679(34) 600(30) 19- - 17+

226.8 -0 .2 3 (7 ) 16(2) 14(2) 43- - 41+

231.9 b) 14(3) 1 2 ( 2 ) 4 9 + - 4 7 -

234.3 a) 1206(60) 1336(67) 13+ - 9+

234.5 a) 196(23) 170(20) 25+ - 23"

238.3 0.47(4 ) -0 .17 (6 ) 19(1) 2 0 ( 1 ) 19- - 15“

249 a) d) 19(2)

249.5 a) 33(4) 29(4) 45+ - 43"

261.1 -0 .2 3 (1 )

270.7 b)

277.7 b)

290.4 a)

290.7 a)

294.8 a)

295 a) d)

297.3 - 0 . 1 2 ( 1 )

301.9 a)

302 a) d)

307.7 d)

308.6 - 0 . 2 1 ( 6 )

313 b) d)

320.7 -0 .18 (8 )

333.6 0.28(7 ) 0 . 0 1 ( 1 0 )

347.5 0 . 2 0 ( 1 ) -0 .0 8 (1 )

358.0 0.30(1) - 0 . 1 1 ( 1 )

361.4 0.36(2) -0 .0 9 (3 )

387.4 0.30(2) -0 .1 8 (3 )

395.0 0 .40(2 ) -0 .2 2 (3 )

413.1 a) c)

414.0 a) c)

415.1 a) c)

422.0 0.31(1) -0 .1 3 (1 )

428.7 0.73(4) -0 .45(10)

450.2 f ) 0 .24(2 )

-9 7 -

263(13) 226(11) 15“ - 13+

6 ( 2 ) 5(1) 47- - 45+

45(7) 38(6) 41+ - 39"

196(10) 167(8) 2 9 + . 27"

102(5) 87(4) 37+ - 35"

1000(50) 1000(50) 17+ - 13+

37(3)

188(9) 160(8) 33+ - 31-

138(7) 129(6) 23- - 19“

8 ( 1 )

24(1)

24(1) 2 0 ( 1 ) (15‘ ) - 13+

28(2)

16(1) 13(1) 51- - 49+

69(4) 66(3) 19- - 15'

404(20) 364(18) 21+ - 17+

309(15) 276(14) 27- - 23'

77(4) 69(3) (27- ) - (23-)

74(4) 66(3) (23-) - (19-)

92(5) 81(4) 25+ - 21+

27(6) 24(5) 29+ - 25+

60(15) 52(13) (15-) - ( l i " )

15(8) 13(7) (31-) - (27-)

251(13) 215(11) 31- - 27-

2 2 ( 1 ) 19(1) 33+ _ 29+

30(2) 26(1) 37+ - 33+

-98-

456.5 0.34(1) -0.19(2) 126(6) 107(5) 35- - 31"

458.6 d) -0.25(3) -0.01(5) 55(3)

465.4 0.23(2) -0.16(6) 33(2) 28(1) 41 + - 37 +

476.9 a) 12(3) 10(3) 45+ - 41 +

478.7 a) 37(3) 31(3) 39" - 35"

503.7 f) 0.41(8) 12(1 ) 10(1) 49 + - 45+

505.0 f) -0 .0 1(6) 16(1) 14(1) 43- - 39-

520.7 0 .02(10) -0.11(19) 6(1) 5(1) 47- - 43'

530.8 d) e) <3 (53+) - 49+

552.4 -0 .02(12) 0.44(32) 5(2) 4(1) 51" - 47-

584.7 d) e) <3

625.9 d) -0.27(4) -0.03(7) 37(2)

a) Doublet in 219Ra

b) Doublet in 220Ra

c) Doublet in 218Ra

d) Multipolarity unknown

e) Doublet with unassigned gamma ray

f) Only three angles available because of problems fitting this peak;

therefore, only second order Legendre polynomial used.

g) Energies are accurate to 0.2 keV

h) Bare gamma ray intensity relative to 294.8 keV gamma ray

i) Intensity (including internal conversion) relative to 294.8 keV

transition.

T a b le 4 .5 B (E 1 : J -» J-1 )/B (E 2 : J-»J-2) V a lu e s in 219Ra The s p in and p a r i t y o f th e i n i t i a l s ta te a re g iv e n by J R e s u lts a re c a lc u la te d from e q u a tio n ( 4 . 2 . 1 ) .

2 x (J 1T- J g s ) B (E 1 ) /B (E 2 ) ( H r 6 fm -2

5 " 2 .5 2 (1 8 )6+ 1 .1 2 (0 8 )7 ‘ 1 .4 9 (1 0 )8+ 1 .2 3 (1 6 )9 " 1 .1 6 (0 8 )10+ 2 .7 7 (6 4 )11- 1 .4 1 (9 )12+ 3 .6 8 (2 6 )13- 2 .1 4 (3 0 )14+ 1 .9 6 (1 4 )15- 1 .7 6 (1 8 )16+ 1 .0 6 (1 7 )17- 2 .0 8 (2 8 )18+ 3 .3 4 (4 8 )19- 1 .3 4 (4 2 )20+ 2 .3 8 (4 4 )21" 4.01(94)

Average 2.09(9)

5 DISCUSSION

I n t h i s c h a p te r , we use o u r s p e c tro s c o p ic in fo r m a t io n to i n v e s t i g a te th e r o le t h a t r e f le c t io n asym m etric shapes p la y i n n u c le a r b e h a v io r i n o u r mass re g io n . S e c t io n 5 .1 p re s e n ts a g e n e ra l d is c u s s io n o f th e s y s te m a tic b e h a v io r o f ene rg y le v e ls and e le c tro m a g n e tic t r a n s i t io n s in th e R n -R a -Th is o to p e s . In s e c t io n 5 .2 , th e e m p ir ic a l e v id e n c e f o r th e a lp h a p a r t ic le c lu s t e r model in th e R a-Th re g io n i s exam ined . T h is i s fo l lo w e d by an e x p l i c i t a lp h a p a r t ic le c lu s te r h y b r id m odel c a lc u la t io n o f th e e n e rg y le v e ls o f a s e r ie s o f Ra is o to p e s i n s e c t io n 5 .3 . The r e la t io n s h ip o f t h e o r e t ic a l p r e d ic t io n s o f th e s t a t ic o c tu p o le d e fo rm at io n m odel to o u r r e s u l t s on 220Ra i s e x p lo re d in s e c t io n 5 .4 .

As we m e n tio n e d in c h a p te r 1 , low ene rg y c o l le c t iv e n e g a t iv e p a r i t y s ta te s have been ob served in m ost re g io n s • o f th e p e r io d ic t a b le ; th e y have g e n e r a l ly been in te r p r e te d i n te rm s o f o c tu p o le v ib r a t io n a l behavi o r . In s e c t io n 5 .5 , we make a d e ta i le d com parison o f s p e c tro s c o p ic in fo r m a t io n on one such re g io n , th e Nd, Sm, Gd and Dy is o to p e s w i th N=82-88, w i t h th e c o rre s p o n d in g in fo r m a t io n on Ra and Th is o to p e s . In s e c t io n 5 .6 , a much b ro a d e r re g io n t h a t in c lu d e s th o se re g io n s m e n tio n e d above i s exam ined v ia th e s y s te m a tic b e h a v io r o f lo w - ly in g n e g a t iv e p a r i t y s ta te s .

F i n a l l y , th e p a r t ic le - c o r e c o u p lin g i n 219Ra and i t s odd-A n e ig h b o rs i s d isc u sse d in s e c t io n 5 .7 .

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5 . 1 S Y S T E M A T I C B E H A V I O R O F E V E N R n , R a a n d T h N U C L E I

B o th 216Rn and 220Ra a re lo c a te d in a n e ig h b o rh o o d in w h ic h a shape t r a n s i t io n fro m s p h e r ic a l (n e a r th e d o u b ly magic 208Pb n u c le u s ) to quad ru p o le deform ed ( f o r exam ple, 232Th) i s ta k in g p la c e . F ig u re 5 .1 shows t h i s t r a n s i t io n d e v e lo p in g as n e u tro n s a re added to th e sem i-m ag ic (N=126) n u c le u s 214Ra. T h is n u c le u s d is p la y s th e i r r e g u la r gamma-ray sequence c h a r a c te r is t ic o f a s in g le p a r t ic le e x c i t a t io n sp ec trum . The spec trum o f y r a s t s ta te s o f s p in s 2f i to 8R can be rep rod uced in th es h e l l m odel by th e b re a k in g o f a p a ir o f h . p ro to n s . As th e n e u tro n9/ 2number reaches 130 ( i n 218R a ), th e e n e rg y le v e l spectrum r e f le c t s a v ib r a t io n a l b e h a v io r . F o r th e e n t i r e p o s i t iv e p a r i t y y r a s t band, as w e l l as f o r n e g a tiv e p a r i t y s ta te s o f s p in g re a te r th a n l i b , th e energy le v e ls a re n e a r ly e q u a lly spaced (As we " w i l l d is c u s s b e lo w , n e g a tiv e p a r i t y s ta te s o f s p in o f H R and le s s seem to have a d i f f e r e n t c h a ra c t e r ) . I n 220Ra, th e y r a s t sequence c o n tin u e s i t s t r a n s i t io n to w a rd a r o t a t io n a l

E t = A J( J + l )

p a t te r n . F o r an id e a l J (J + 1 ) p a t te r n , th e r a t i o E (4 +) / E ( 2 +) = 3 .3 3 ; in th e l i m i t o f a ha rm on ic v ib r a t o r , E (4 +) / E ( 2 +) = 2 .0 0 . In 218Ra and 220Ra, t h i s r a t i o ta k e s th e v a lu e s 1 .9 0 and 2 .3 0 , r e s p e c t iv e ly . Thus, 220Ra i s s t i l l c lo s e r to th e v ib r a t io n a l l i m i t th a n to th e r o t a t io n a l one. The E (4 +) /E ( 2 +) r a t i o c o n tin u e s to in c re a s e as n e u tro n s a re added,

re a c h in g 3 .2 1 in 228Ra.T h is t r a n s i t io n from s p h e r ic a l to deform ed shape i s f u r t h e r

F ig u re 5 .1 Energ y le v e l s y s te m a t- ic s o f A=214-222 Ra even-A is o to p e s . The decrease i n th e e x c i t a t io n energy o f th e f i r s t e x c ite d 2+ s ta te w ith in c re a s in g n e u tro n number demons t r a te s th e o n s e t o f q u a d ru p o le •c o l l e c t i v i t y . I n b o th 214Ra and 216Ra, th e lo w e s t - ly in g n e g a tiv e p a r i t y s ta te i s th e two q u a s ip a r t ic le 11- s t a te . A band o f n e g a tiv e p a r i t y s ta te s a t lo w e r e x c i t a t io n energy appears in th e h e a v ie r is o to p e s . The d a ta a re ta k e n fro m (Ga 83b, Go 85, Le 78 , Ho 79 , Ch 82) and th e p re s e n t w o rk .

LEVEL STRUCTURE OF Ra NUCLEI

3 3 9 0 - ■17" 3389-

2l6Rn 88™ 128 2l8Rn8 8 Ka

0 1 ^ 0 *

(I4+) 2 9 6 0 -

(13") 2 6 8 8 - j

2 5 2 1 '

2 2 6 0 '

2 1 0 3 -

1 8 6 2 -

1710-

I4 § 4 "

I 3 4 1 -

I 1 6 2 -

1000-8 7 2 -

6 8 7 . 6 3 3 - 4 7 4 . 4 1 2 , 4 0 9

178-

0-

- r I 8 +

4- 1 7 "

T I 6 +

1 5 "14+

1 3 "

I2+

i r i o + 9 "

8 + 7 "

. 6 +

£42+0 +

1 0 2 5 -9 1 4 -

4 7 3 - . 3 1 7 . 3 0 l-I 2 4 2

I I I' O’

■(2+)•(0+)

4r2+

re f le e te d i n th e d e c lin e o f th e e n e rg ie s o f th e f i r s t e x c ite d 2+ s ta te s w i t h in c re a s in g n e u tro n num ber. The f ig u r e shows is o to p e s o n ly to mass 222 , f o r w h ich E (2 +) = l l l keV . However, t h i s t re n d a ls o c o n tin u e s to 228Ra [E (2 + ) = 64 keV] .

Even more s t r i k i n g i s th e sudden appearance o f low to medium s p in ( J < llT i) n e g a tiv e p a r i t y s ta te s w i th th e a d d it io n o f two n e u tro n s to 216Ra. A J ^ l l " s ta te i s p re s e n t in b o th 214Ra and 216Ra; how ever, b o th s ta te s can be u n d e rs to o d in te rm s o f s in g le p a r t ic le e x c i ta t io n s (Ho 79, I t 8 3 ) . On th e o th e r hand, th e J c l lh n e g a tiv e p a r i t y s ta te s found in 218Ra a re a t e n e rg ie s f a r to o low to be e x p la in e d in t h i s way. In s te a d , th e y m ust be a s s o c ia te d w i th a c o l le c t iv e phenomenon (Sh 8 5 ) .

The J C llT i n e g a tiv e p a r i t y s ta te s i n 218Ra show a c u r io u s d if fe re n c e fro m th e p o s i t iv e p a r i t y s ta te s i n th e same s p in ra n g e . Even though th e p o s i t iv e p a r i t y s ta te s con fo rm to a v ib r a t io n a l p a t te r n , th e n e g a tiv e p a r i t y s ta te s e x h ib i t a c o n s id e ra b ly more r o t a t io n a l b e h a v io r . T h is is d e m o n s tra te d in f ig u r e 5 .2 , a p lo t o f a lig n e d a n g u la r momentum [ J ( x ) = J + l/2 f o r a K=0 band] as a fu n c t io n o f Tiw, w h ich i s g iv e n b y :

hu» = ( 1 / 2 ) ( E (J + l) - E ( J - l ) }

In th e p o s i t iv e p a r i t y band, Tiw i s c o n s ta n t ; in c o n t r a s t , "hw in c re a s e s w i th s p in in th e n e g a tiv e p a r i t y band.

An a b ru p t change occu rs in th e n a tu re o f th e 218Ra n e g a tiv e p a r i t y band a t J = l lh . T h is change, c le a r ly v i s ib le in f ig u r e 5 .2 , l i k e l y i n d i c a te s a c o n s id e ra b le a d m ix tu re o f tw o - q u a s ip a r t ic le com ponents, s im i la r to th o se re s p o n s ib le f o r th e 11 " s ta te i n 216Ra. A few t e n t a t i v e ly a ss ig n e d le v e ls in a s id e band o f 218Ra seem to c o n tin u e th e d o m in a n tly

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F ig u re 5 .2 I ( x ) v s . Rw p lo t f o r 218Ra and 220Ra. Dashed l in e s d enote t e n t a t i v e ly a ss ig n e d s ta te s . Data a re ta k e n fro m (Ga 83b, Go 85) and th e p re s e n t w o rk .

-104-

Such a band c ro s s in g does n o t seem to o c cu r in 220Ra (see f ig u r e 5 . 2 ) . In s te a d , th e two o p p o s ite p a r i t y bands appear more a l i k e and show no sudden changes. However, i t sh o u ld s t i l l be n o te d t h a t a t low s p in s th e n e g a tiv e p a r i t y band i s somewhat more r o t a t io n a l in c h a ra c te r th a n th e p o s i t iv e p a r i t y band in th e sense t h a t th e n e g a t iv e p a r i t y ene rg y le v e ls a re c lo s e r to th e J (J + 1 ) p a t te r n .

I t seems u n l i k e ly t h a t th e c o l le c t iv e n e g a t iv e p a r i t y s ta te s th a t a re so p ro m in e n t in 218Ra and 220Ra do n o t e x is t a t a l l in 216Ra. In s te a d , we sug g est t h a t th e se s ta te s , w h ic h decrease i n ene rg y from 218Ra to 220Ra, s im p ly become n o n -y ra s t i n 216Ra. Y r a s t s ta te s a re p r e f e r e n t ia l l y p o p u la te d i n heavy io n fu s io n -e v a p o ra t io n re a c t io n s such as th e ones used to s tu d y h ig h s p in s ta te s in th e Ra is o to p e s , as n o te d p r e v io u s ly ; th u s , n o n -y ra s t s ta te s w ou ld be o n ly w e a k ly p o p u la te d i n th e (H I ,x n ) re a c t io n s used to s tu d y these is o to p e s .

F ig u re 5 .3 shows th e energy le v e l s y s te m a tic s o f th e Th is o to p e s o f masses 218 to 226 . The im p o r ta n t fe a tu re s o f th e Ra s y s te m a t ic s , the s p h e r ic a l to deform ed t r a n s i t io n and th e appearance o f lo w -s p in n e g a tiv e p a r i t y s ta te s , a re a p p a re n t i n th e Th is o to p e s a ls o . The is o to n ic p a r t n e rs o f th e s e two e lem en ts a re s t r i k i n g l y a l i k e i n s t r u c tu r e . T h is s im i l a r i t y i s somewhat s u r p r is in g in v ie w o f th e w e ll-k n o w n im p o rta n c e o f th e p ro to n -n e u tro n in t e r a c t io n in c a u s in g q uad rup o le d e fo rm a tio n (Ta

6 2 ) .We compare 216Rn to th e n e ig h b o r in g N=130 is o to n e s and th e Rn is o

to p e s i n f ig u r e 5 .4 . The is o to n ic c h a in d is p la y s a g ra d u a l o n s e t o f q u a d ru p o le d e fo rm a tio n w i th in c re a s in g p ro to n num ber. T h is e f f e c t

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collective negative parity band to higher spins (Go 85).

F ig u re 5 .3 Energy le v e l s y s te m a t- ic s o f A=218-226 even-m ass Th is o to p e s . These is o to p e s d is p la y c h a ra c t e r is t ic s v e ry s im i la r to th o se seen in th e Ra is o to p e s ( f ig u r e 5 .1 ) . The d a ta a re ta ke n from (Bo 85 , Wa 8 3 , Bo 83b, Bu 8 5 ) .

LEVEL STRUCTURE OF Th NUCLEI

3072---------(15")

2106 I0H

195'

690-

2609---------(13")

2875--------17“2691------- 16+

2434------- 15"2262------- 14+2159-------11 “

2013-------10+ 2017--------13“1853------- I2 +

8 1719---------9“,565--------6+ 1598------- 8+ 1624--------11“

1462--------I0+

1256-------- 9“1094------- 8+

924------- 7"

1329-------- 7 "I 166---------6 +99 4 -------- 5"

373

760 ------- 4+ 750 6 +651--------- 5“

4 4 0 ---------4+

183--------- 2+0 0 + 0 0 + 0 0 +

1553-------- (12+)1345- ---- (II")1180- -----(I0+)9 9 8 - -----(9 " )833 —-----8 +697 —---- 7 "532 6 +462 _ ___ 5"316 (3")280 —-----4 +246 1"9 3 “ ----- 2 +

0 “ 0 +

2861 — ---- 19"

2635— — 18+

2413— ----17“

2196 — — 16+

1989 — — 15"

1782 — — 14+1596 — — 13"

1395 — — I2 +1238 — — I I "

1041 — — I0 +923 — — 9"722____8+658 — — 7"45 L ,5"448 = = 6 +307.__— -3"2 30 ,= = |“2 2 / s4+

7 2 ; = — x2 +0 0 +

* 9 0 ™ ,28 2|gT h l30 2 l| lh I32 2|4 Th,,4 22®Th9 0 ' "134 90 1 n 136 gi

F ig u re 5 .4 Energ y le v e l s y s te m a t-ic s o f N=130 is o to n e s (Z>82) and Rn is o to p e s (A > 212). The is o to n e s 218Ra and 220Th d is p la y s t r i k i n g l y s im i la r s t r u c tu r e . The d a ta a re ta k e n from(Ku 76 , Ku 77 , Pe 81 , Go 81 , Ho 79, Ga 83b, Bo 85 , Le 78) and th e p re s e n t w o rk .

N = 130 IS0T0NES

(8 )_ (6 )_ ( 4 + ) ‘

(2+)-

1 3 3 5

’ 1 2 7 7

1 1 1 7

- 8 0 6

0+ 02 1 2 P b

8 2 k d I 3 0

( I O + ) - r ~ 2 1 1 1 . 5

! 4 6 5 . 9 ( 1 3 )

(8+ )

(4+)■ 6 0 9 .

0* 02 1 4 P n 8 4 k o I 3 0

• 1 6 4 5 . 6

4 1 9 . 4 ( 5 7 )

( 6 + J - r 1 — 1 2 2 6 . 2

3 8 5 . 4 ( 6 0 )

8 4 0 . 8

1 3 7 8 . 9 ( 6 8 )

6 1 . 9 _________

4 6 1 . 9 ( 1 0 0 )

- 02 1 6 R n

8 6 K n l 3 0 9 0 T h l 3 0

I Q - 2 6 3 2

0+ 0212 Rn

8 6 1 2 6

(i r

o+— o2 l 4 Rn

8 6 1 2 8

Rn ISOTOPES- 2 2 5 8

1 0 * ----------- 1 7 8 7

( 8 + )8 1 —

1 6 2 5

6 ----------- 1 4 4 2

4 +— 1 1 4 0( 6 +y

( 4 + )V — ■ 6 9 ^

" 2 + '

( I O ^ ) - i— 2 1 1 1 . 5

1 4 6 5 . 9 ( 1 3 )

A - 1 6 4 5 . 6

4 1 9 . 4 ( 5 7 )

- 1 2 2 6 . 2

3 8 5 . 4 ( 6 0 ) f

p 8 4 0 . 8 ( V =

3 7 8 . 9 ( 6 8 ) ( 4 + ) ---------6 5 3

+ 6 1 . 9 . _______ 4. 2 * ------------ 3 2 4 —

4 6 1 . 9 ( 1 0 0 )

- 0 0+--- 02 1 6 R n 2 1 8 R n

8 6 1 3 0 8 6 1 3 2

8 4 0

: 7 9 7

( 4 + )534(3 ) ;

—54( D: 6 6 3

6 4 5

- 2 - -------------- 2 4 1

0+ 02 2 0 Rn

8 6 1 3 4

appears to s a tu ra te a t Z=88 s in c e , as n o te d above, 220Th lo o k s no more d eform ed th a n does 218Ra. H ow ever, th e o u ts ta n d in g fe a tu re in th e is o t o n ic c h a in i s th e appearance o f th e lo w - ly in g n e g a tiv e p a r i t y s ta te s in 218Ra. Once a g a in , we can su g g e s t t h a t th e c o rre s p o n d in g n e g a tiv e p a r i t y s ta te s in 216Rn a re n o n -y ra s t and, c o n s e q u e n tly , unobserved in o u r fu s io n -e v a p o ra t io n r e a c t io n . T h is b e h a v io r w ou ld be c o n s is te n t w i th th e o b s e rv a t io n t h a t th e 1 " and 3 " s ta te s in th e even A=218-222 Rn is o to p e s a re c o n s id e ra b ly h ig h e r in e n e rg y th a n th o se in Ra and Th f o r N>132 (s e e , f o r exam ple, f ig u r e 1 . 6) .

The Rn is o to p ic c h a in shows, as do th e Ra and Th c h a in s , a more p e r s is t e n t o n s e t o f q uad rup o le d e fo rm a t io n th a n t h a t seen in th e is o t o n ic c h a in . As n o te d in th e in t r o d u c t io n , th e r e la t i v e p o s i t io n s o f th e 1 " and 3 “ s ta te s i n 218Rn, 220Rn and 222Rn a re in te r e s t in g in th e c o n te x t o f c o rre s p o n d in g d a ta on Ra and T h . In 220Rn and 222Rn, th e t e n t a t i v e ly a ss ig n e d 1 " s ta te s f a l l 18 keV and 35 keV, r e s p e c t iv e ly , b e low th e 3 " s ta te s . The ( I - ) s ta te i n 218Rn a c t u a l ly l i e s 43 keV above th e ( 3 ) " s t a te . In a l l Ra and Th is o to p e s in w h ic h b o th th e 1 " and 3 " s ta te s a re known, th e 1 " s ta te l i e s a t le a s t 60 keV be low th e 3 " s ta te .

We now tu r n o u r a t t e n t io n to th e s y s te m a tic b e h a v io r o f e le c tro m a g n e t ic t r a n s i t io n s . In p a r t ic u la r , we a re in te r e s te d in th e b e h a v io r o f E l d e e x c ita t io n s . The m ost d i r e c t way o f g a in in g in fo rm a t io n on e le c tro m a g n e tic t r a n s i t io n m a t r ix e le m e n ts i s by a measurement o f th e l i f e t im e s o f e x c ite d s ta te s based, f o r exam ple, on th e D o p p le r - s h i f t in g o f gamma ra y s e m it te d fro m r e c o i l in g re s id u e s (Fo 7 4 ) . Such a m easurem ent i s im p r a c t ic a l f o r 220Ra f o r two re a s o n s . F i r s t , th e 208P b (14C ,2 n )220Ra r e a c t io n does n o t d e l iv e r enough r e c o i l v e lo c i t y to th e r e s id u a l n u c le u s

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to s e p a ra te s u f f i c i e n t l y th e D o p p le r - s h if te d gamma-ray peaks fromn o n - s h i f te d p eaks . Second, an in v e rs e r e a c t io n , 14C (208P b ,2 n )220Ra,

w ou ld re q u ire a r a d io a c t iv e 14C ta r g e t , w h ic h was n o t r e a d i ly a v a i la b le .In l i e u o f d i r e c t m easurem ents o f m a t r ix e le m e n ts , we tu r n to th e

B (E 1 : J -» J -1 ) /B (E 2 : J-+J-2) r a t i o , w h ic h can be e x t ra c te d fro m th e gamma ra y b ra n c h in g r a t io s (see c h a p te r 4 ) . T h is r a t i o i s u s e fu l because r e l ia b le e s t im a te s o f B (E 2) v a lu e s th ro u g h o u t th e p e r io d ic ta b le can be madeu s in g e x p re s s io n s such as th o se g iv e n by G ro d z in s (G r 6 2 ) . An a l t e r n a t i v e e x p re s s io n t h a t p re d ic ts B (E 2) v a lu e s more a c c u ra te ly f o r o u r immed ia te re g io n i s t h a t suggested i n (Bo 8 5 ) , n am e ly :

T 1 /2 /s e c = 0 .9 0 8 {E y (2 +->0+) / k e V } “ 4 - ° 6/ ( l+ a t o t ) ( 5 .1 .1 )

where a,. .. i s th e e le c t r o n c o n v e rs io n c o e f f ic ie n t f o r th e 2+-»0+ t r a n - t o ts i t i o n . From t h i s e x p re s s io n f o r th e h a l f - l i f e , we can d e r iv e o u r e s t i mate f o r B (E 2 :2 +-»0+) , nam e ly:

-0 94B (E 2 :2 +-»0+) = (6 .2 4 x 10 5) (E /k e V ) (e 2fm 4) ( 5 .1 .2 )

In o rd e r to s tu d y b ra n c h in g r a t io s f o r gamma d e e x c ita t io n s o f h ig h s p in s ta te s , we m ust be a b le to make e s t im a te s o f B (E 2 ) v a lu e s f o r these t r a n s i t io n s fro m th e e s t im a te f o r th e 2+-»0+ t r a n s i t io n , e q u a tio n ( 5 . 1 . 2 ) . An id e a l r o t o r has a p a r t i c u la r l y c o n v e n ie n t e x p re s s io n f o r B (E 2 :J -» J -2 ) in te rm s o f th e i n t r i n s i c e le c t r ic quad rup o le moment, Qq , nam ely:

B(E2 : J-»J-2) = (1 5 /3 2 n )e 2Q0 2J . ( J - l ) / ( 4 J 2- l ) ( 5 .1 .3 )

(Bo 7 5 ) . I n te rm s o f B (E 2 :2 +-»0+) , t h i s i s

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- 1 1 0 -

Even though th e e x p re s s io n ( 5 .1 .4 ) assumes a p e r fe c t r o t o r , th e l i f e

tim e s o f th e 2+ and 4+ s ta te s i n 222Th a re w e l l rep rod uced by i t (Bo8 5 ) . T hus , we use i t w i th c o n fid e n c e to make rough e s t im a te s f o r h ig h s p in s ta te d e e x c ita t io n s in th e Ra is o to p e s .

A c o n v e n ie n t y a r d s t ic k f o r e le c tro m a g n e tic t r a n s i t io n s t re n g th i s th e W e issko p f s in g le p a r t ic le u n i t ( W .u . ) . The W .u. rem oves, in a conv e n ie n t b u t h ig h ly a p p ro x im a te way, th e mass dependence o f reducedm a t r ix e le m e n ts and a llo w s com p arison o f t r a n s i t i o n s tre n g th s fro m d i f f e r e n t re g io n s o f th e p e r io d ic ta b le . An E2 t r a n s i t io n o f 1 -5 W .u. i s c o n s is te n t w i th th e s t r e n g th exp ec ted in a s in g le p a r t ic le t r a n s i t io n .In deform ed r o to r s , how ever, B (E 2) v a lu e s may reach 250 W .u . S in g lep a r t ic le E l t r a n s i t io n s i n heavy (A>50) n u c le i , on th e o th e r hand, gene r a l l y have B (E 1) v a lu e s le s s th a n 10" 4 W .u.

The W .u . i s d e f in e d f o r E l and E2 t r a n s i t io n s as

B, ( E1) = 6.446x10"2xA2Z3 e 2fm 2 w

and,

BW(E2) = 5 .9 4 0 x 1 0 -2xA4 /3 e2fm 4

(Bo 6 9 ) . C o n se q u e n tly , th e e x p re s s io n f o r B (E 1 : J -» J -1 ) /B (E 2 : J-»J-2) can be w r i t t e n i n te rm s o f W .u . as

B(E2 : J-»J-2) = (15/2)B(E2 :2+->0+) J(J-1)/(4J2-1) (5.1.4)

B (E 1 ) /B (E 2 ) = { l y ( E l ) / I y ( E 2 ) } x{ (EE2/M e V )V (E E1/M e V )3 }x (7 . 1 0 x lQ -7 )A 2/3 ( 5 .1 .5 )

( B ( E l) /B ( E 2 ) } ( W .u . ) = { B ( E l) /B ( E 2 ) } ( fm - 2 )x (0 .9 2 1 5 A2 /3 ) ( 5 .1 .6 )

F ig u re 5 .5 shows B (E 1 : J -» J -1 ) /B (E 2 : J-»J-2) r a t io s f o r th re e Ra is o to p e s . V a lu e s f o r 220Ra a re a f a c to r o f 2 .5 lo w e r th a n th o s e o f 218Ra. In o rd e r th e n to deduce th e b e h a v io r o f th e E l m a t r ix e le m e n ts , we m ust exam ine E2 m a t r ix e le m e n ts f o r th e se tw o n u c le i . In a re c e n t m easurement o f l i f e t im e s in 218Ra fro m t h i s L a b o ra to ry (Ga 86) , E l and E2 m a t r ix e le m e n ts f o r d e e x c i ta t io n o f s ta te s o f J>4Ti were d e te rm in e d to be n e a r 6 x l0 -3 W .u. and 60 W .u ., r e s p e c t iv e ly . E x p re s s io n ( 5 .1 .2 ) y ie ld s a B (E 2) o f 60 W .u. f o r th e 2+->0+ t r a n s i t io n in 220Ra; t h i s t r a n s la te s by ( 5 .1 .4 ) to 100 W .u. f o r J=8 . I f we c o n s id e r th e average v a lu e o f B (E l) /B (E 2 )= 4 x lO ~5 W .u . ( 1 . 2 x l0 -6 fm '2 ) f o r 220Ra, th e n we c a lc u la te an e s t im a te o f B (E l)= 4 x lO ' 3 W .u. a t J=8 i n t h i s n u c le u s . We conc lud e t h a t th e ob served decrease in B (E 1 ) /B (E 2 ) r e f le c t s a decrease i n E l enhancem ent.

In fo r m a t io n re g a rd in g 226Ra a v a i la b le in ( Z i 80) g iv e s B (E 1 )/B (E 2 ) r a t io s a f a c to r o f te n lo w e r th a n th o se f o r 218Ra. W ith an e s t im a te o f 240 W .u . f o r B (E 2) a t J=8 , we f in d th a t B(E1 )= 1 . 5 x l0 -3 W .u. These B (E 1 )/B (E 2 ) v a lu e s appear to r e f l e c t a sm ooth, m o no ton ic decrease w ith n e u tro n num ber.

T h is sm ooth f u n c t io n a l dependence i s even more a p p a re n t in th e Th is o to p e s (see f ig u r e 5 .6 ) . D ata on gamma ra y b ra n c h in g r a t io s a re a v a i la b le f o r f i v e such is o to p e s ra n g in g in n e u tro n number fro m 130 to 142; t h i s in fo r m a t io n shows a smooth decrease o f th re e o rd e rs o f m agnitude in th e B (E 1 ) /B (E 2 ) r a t i o as th e n e u tro n number in c re a s e s .

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We can also convert between the two units by

Figure 5.5 B(E1)/B(E2) ratios forthree even-even Ra isotopes. The data are taken from (Ga 83b, Zi 80) and the present work.

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F ig u re 5 .6 B (E 1 )/B (E 2 ) r a t io s f o rf i v e e ve n -e ve n Th is o to p e s . The d a ta a re ta k e n fro m (Bo 85 , Wa 83 , Bu 85, Ha 84 , Le 8 0 b ).

o .CD

O -

c_

B ( E I ) / B ( E 2 ) ( f m )- 2

o .0 3

o .-si

o ,<T>

1 1 1 Mil 1 1 1 Mi l l

• o , 1 O ,

-nI CD

r o r ocu ► r o «r o 0 0

"H HZT I T

TTTTTTT

t oO ,CJi

O ,CJl

- e i i -

5 . 2 I N T E R P R E T A T I O N O F R a A N D T h I S O T O P E S I N T E R M S O F A N

A L P H A P A R T I C L E C L U S T E R M O D E L

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In t h i s s e c t io n , we summarize th e e x p e r im e n ta l ev id ence f o r a lp h a p a r t ic le c lu s te r in g in even is o to p e s o f Ra and T h . We w i l l in c lu d eth re e e x p e r im e n ta l o b s e rv a b le s in o u r d is c u s s io n : ground s ta te a lp h ap a r t ic le decay w id th s , B (E 1 )/B (E 2 ) r a t io s and th e a lp h a p a r t ic le decay h in d ra n c e fa c to r s f o r th e I - and 3 " s ta te s . H a v in g p r e v io u s ly i n t r o duced th e f i r s t two q u a n t i t ie s , we now d is c u ss th e t h i r d .

The r a te o f a lp h a p a r t ic le decay to a s ta te i s a f fe c te d by th re e fa c to r s - th e Q -v a lu e o f th e decay, th e d i f fe r e n c e in s p in betw een th e p a re n t and d a u g h te r s ta te s , and the r e la t io n s h ip o f th e i n t r i n s i c s t r u c tu re s o f p a re n t and d a u g h te r s ta te s . The h in d ra n c e f a c to r i s computed in such a way as to u n fo ld th e Q -va lu e dependence fro m an a lp h a p a r t ic le decay p a r t i a l h a l f - l i f e . More p r e c is e ly , th e h in d ra n c e f a c to r o f an a lp h a p a r t ic le decay t r a n s i t io n from th e g round s ta te o f an even -even n u c le u s i s d e f in e d to be th e q u o t ie n t o f th e e x p e r im e n ta l p a r t i a l h a l f - l i f e o f a s p e c i f ic t r a n s i t io n and the h a l f - l i f e computed f o r some chosen fo r m u la t io n o f a lp h a p a r t ic le decay th e o ry , assum ing th a t th e d a u g h te r s ta te has J ir=0+ (Hy 6 4 ) . V a lu e s f o r Z=86-94 can be found in f ig u r e 5 .7 . These v a lu e s a re ta k e n fro m (Le 7 8 ) , and r e f l e c t th e cho ice o f a lp h a p a r t ic le decay th e o ry made in th a t w o rk . As we have s ta te d , th e s p in dependence has been l e f t in th e a lp ha p a r t ic le h in d ra n c e fa c to r s shown;how ever, t h i s has a r e l a t i v e l y s m a ll e f f e c t , e s p e c ia l ly f o r J ^ l " s ta te s( f o r d is c u s s io n o f th e s p in dependence, see (Hy 6 4 ) ) .

The systematic behavior of the alpha particle hindrance factors

Figure 5.7 Alpha p a r t ic le hindrance factors for Z=86-94. The abscissa i s the valence neutron number. The data are taken from (Le 78).

I

seem to in d ic a te t h a t th e s t r u c tu r e o f th e 1 " s ta te s and g round s ta te s a re v e ry s im i la r f o r th e l ig h t e r is o to p e s in th e re g io n (Ra and Th n e a r N=132, and Rn is o to p e s ) . To r e s ta te t h i s , th e im p o r ta n t components o f th e shapes o f 1 ' s ta te s in th e se l ig h t e r n u c le i a re a ls o im p o r ta n t comp o n e n ts o f th e g round s ta te shapes.

The e x p e r im e n ta l argum ent f o r a lp h a p a r t ic le c lu s te r in g in Ra andTh is o to p e s can be s ta te d in th e f o l lo w in g way (w h ic h i s i l l u s t r a t e d in f ig u r e 5 . 8 ) . The la r g e s t ground s ta te a lp h a p a r t ic le decay w id th s , s m a lle s t a lp h a p a r t ic le h in d ra n c e fa c to r s f o r 1“ and 3 " s ta te s , and la r g e s t B (E 1 ) /B (E 2 ) r a t io s a l l occur a t N=130 and N=132. As n e u tro n s a re added, th e a lp h a p a r t ic le decay w id th s d e c rease , h in d ra n c e fa c to r s in c re a s e and B (E 1 ) /B (E 2 ) r a t io s d ec rease . In s h o r t , th e tre n d s i n these th re e q u a n t i t ie s appear to be c o r r e la te d and th u s to r e f l e c t th e same n u c le a r phenomenon; th e la rg e a lp ha p a r t ic le decay w id th s sug g est t h a t t h i s phenomenon to be t h a t o f a lp ha p a r t ic le c lu s te r in g .

The q u e s t io n o f w h e th e r such b e h a v io r , in c lu d in g th e la rg e a lp h ap a r t ic le w id th s , can a ls o r e f l e c t a s t a t i c o c tu p o le shape rem a ins open.

5 . 3 S T U D Y O F E V E N R a I S O T O P E S T H R O U G H C A L C U L A T I O N S W I T H

T H E U ( 6 ) ® U ( 4 ) H Y B R I D M O D E L

As shown in c h a p te r 2 , a s u b s ta n t ia l number o f f r e e p a ra m e te rs a re a v a i la b le i n th e h y b r id m odel. I t i s d e s ir a b le , how eve r, to use t h i s

model in a way such t h a t th e sp e c tro sc o p y o f s e v e ra l is o to p e s can be u n d e rs to o d u s in g few o f th e se ind ep end en t , degrees o f freed om . Here we

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F ig u re 5 .8 The s y s te m a tic b e h a v io r o f g round s ta te reduced a lp h a p a r t i c le decay w id th s , a lp h a p a r t ic le decay h in d ra n c e fa c to r s and B (E 1 ) /B (E 2 ) r a t i o s . The a b s c iss a is th e n e u tro n number. The d a ta a re ta k e n from (Le 78 , Bo 85 , Wa 83 , Bu 85 , Ha 84 , Le 80b, Ga 83b, Z i 80 , Ra86) and th e p re s e n t w o rk . The ground s ta te a lp h a p a r t ic le decay w id th s were c a lc u la te d by th e method o f G ai (Ga 86) , w h ich adds a q uad rup o le d e fo rm a tio n to th e method o f Rasmussen (Ra 5 9 ) .

B (El)/

B(E2

)(fm

2)-117-

p re s e n t a c a lc u la t io n o f th e e x c i t a t io n s p e c tra o f f o u r Ra is o to p e s (22 0- 226£a ) i n wh ic h th e v a lu e s o f a l l b u t one p a ra m e te r have been

f ix e d . P a ra m e te r v a lu e s used a re shown in ta b le 5 .1 .T h is c a lc u la t io n was p e rfo rm e d by f i r s t v a ry in g a l l p a ra m e te rs to

f i n d a good f i t to th e d a ta f o r 220Ra. F o r th e c a lc u la t io n s p e r ta in in g to th e re m a in in g th re e is o to p e s o n ly £p was a d ju s te d . The r e s u l t in g c a lc u la te d s p e c tra a re shown in f ig u r e 5 .1 0 . In g e n e ra l, y r a s t p o s i t iv e p a r i t y and lo w ene rg y n e g a t iv e p a r i t y s ta te s a re f i t t e d q u ite w e l l . P rob lem s d e ve lo p in th e c a lc u la t io n s , how ever, f o r 226Ra in th e n o n -y r a s t 0+ and 2+ s ta te s as w e l l as f o r th e l - ,2 - , 3 ' t r i p l e t .

The v a lu e s used f o r and ic a re com parable in m agn itude and, th u s , r e f l e c t th e t r a n s i t i o n a l n a tu re o f th e se Ra is o to p e s . I t i s in t e r e s t in g to n o te t h a t th e c a lc u la te d e x c i t a t io n s p e c tra seem to rep rod uce th e s u b s ta n t ia l v ib r a t io n a l - to -d e fo rm e d change in t h i s s e t o f is o to p e s w ith o u t a c o rre s p o n d in g change in th e r e la t i v e s t re n g th s o f the E^n^ and te rm s . G e n e ra lly , such a t r a n s i t io n i s rep rod uc ed in

th e IBA by v a r y in g th e r a t i o ( Sc 7 8 , Ca 8 5 ) .The one p a ra m e te r t h a t i s v a r ie d , e ( th e n o ta t io n i s t h a t o f (DaP

8 3 ) ) , c o n t r o ls , i n a ro u g h ly l in e a r fa s h io n , th e o f f s e t o f th e 0 + andgsI - s ta te s . The p re c is e p h y s ic a l m eaning o f t h i s p a ra m e te r i s n o t c le a r ; h o w eve r, we m ig h t s p e c u la te t h a t t h is p a ra m e te r depends on th e shape o f th e c o re -a lp h a p a r t ic le p o t e n t ia l in th e f o l lo w in g way. We w o u ld exp ec t t h a t t h i s p o t e n t ia l w ou ld lo o k , i n one d im e n s io n , s im i la r to t h a t i l l u s t r a te d i n f ig u r e 2 .9 f o r th e s t a t ic o c tu p o le d e fo rm a tio n w i t h a f i n i t e b a r r i e r . C o n se q u e n tly , th e doub le o s c i l l a t o r te c h n iq u e used in (Le 82) to s o lv e th e o n e -d im e n s io n a l f i n i t e b a r r ie r p rob lem sh o u ld be e q u a l ly as

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h y b r id model c a lc u la t io n (see t e x t ) compared to E ( l ' ) . The d a ta a re ta k e n fro m (Le 7 8 ) .

Figure 5.9 The parameter ep in the

E(l")

(keV)

€ d

(MeV

)

F ig u re 5 .1 0 Com parison o f ene rg y le v e ls p re d ic te d u s in g h y b r id model (see t e x t ) to observed le v e l s p e c tra . The d a ta a re ta k e n fro m (Ku 76 , Ku 77 , Z i 80) and th e p re s e n t w o rk .

keV

_ ke

V

1000-

800-

600 -

400-

2 0 0 -

° — 1200-

1000-

800-

600-

400 -

2 0 0 -

0 -

1200- E X P T g - 1 1 6 2 T H E O R Y

911130

8* 1000 8* 1012

r 872 7*846

6* 687 6+68l51633 51631

31473 314744* 409 412 4* 404 3?8

2+ 178 2* 181

OLQ 22QBn QU>

E X P T T H E O R Y

2* 1064(2*) 993° l i ! 6 01891

5* 433 5* 411

A* 25!3 -90 —5 LH6

2* 840+ 0 224-

4+2543— 6 I 2152* 94

0+ 0

EXPT THEORY

(2*) 1025 21(0*1914 Q+

5liI 4 51464

4 * 301 T 3 j7 4+ 3Q33~ 325l~ 242 r 245

2* III 2U2.8

^ 222 Roe x p t t h e o r y

II_II33(3-j b 104) , ,-infla ?

912J037

1LL03

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P A R A M E T E R S U S E D I N T H E V I B R O N M O D E L F I T S

S H O W N I N F I G U R E 5 . 1 0

TABLE 5.1

ed 0 .1 8 0 MeV

Kd -0 .0 8 5 MeV

k ’( ARa) 0 .0 0 6 MeV

ic1 ( A“ 4Rn+o) 0 .0036 MeVe 22BRap

0 .0 2 MeV222Ra -0 .1 2 MeV224Ra -0 .1 4 MeV226R3 0 .1 0 MeV

“ P 1 .000 MeV

K -0 .0 3 0 MeV

Aa -0 .2 5 MeV

TS -0 .0 8 0 MeV

X -2 .2 5

s im p l i f ie d p rob lem i l l u s t r a t e s th e dependence o f th e p a r i t y s p l i t t i n g onth e h e ig h t o f th e p o t e n t ia l b a r r ie r s e p a ra t in g th e a lp h a c lu s te r - c o r e

c o n f ig u r a t io n and i t s m ir r o r im age. S in c e gp d e te rm in e s t h i s p a r i t ys p l i t t i n g i n th e h y b r id model c a lc u la t io n , we can deduce t h a t anin c re a s e in z w ou ld r e f l e c t a decrease in th e s iz e o f th e p o t e n t ia l

pb a r r ie r im p lie d in th e h y b r id model c a lc u la t io n . I n th e c a lc u la t io n p re s e n te d h e re , th e v a lu e s o f gp f o l lo w th e t re n d o f th e e n e rg y o f th e 1 " s ta te (see f ig u r e 5 .9 ) .

T h is c a lc u la t io n d e m o n s tra te s t h a t t h i s m odel can in d e e d be used in a way th a t does n o t re q u ire a la rg e number o f f r e e p a ra m e te rs .

\- 1 2 2 -

applicable to the core-alpha particle problem. The solution to this

5 . 4 P R E D I C T I O N S F O R T H E O C T U P O L E D E G R E E O F F R E E D O M I N

22 ° R a

We c o n s id e r , in t h i s s e c t io n , th e sp e c tro sc o p y o f 220Ra i n th e cont e x t o f s e v e ra l c a lc u la t io n s in c lu d in g th e o c tu p o le degree o f freed om . The s tu d y o f (Le 82a) p re d ic ts a s t a t i c o c tu p o le d e fo rm a t io n i n th e g round s ta te o f 220Ra. However, t h i s s tu d y a ls o p re d ic ts a s t a t i c o c tu p o le shape f o r th e ground s ta te o f 218Rn, a p r e d ic t io n w h ic h i s i n cont r a d ic t io n to th e observed o rd e r in g o f th e ( 3 ) - s ta te b e lo w th e ( I - ) s ta te in t h i s n u c le u s . F o r th e case o f a s t a t i c a l l y o c tu p o le deform ed n u c le u s , th e 1~ s ta te w ou ld be lo c a te d b e low th e 3 “ s ta t e ; th e observed c o n f ig u r a t io n i s , in s te a d , c o n s is te n t w i t h th e c o u p lin g o f an o c tu p o le phonon to a n e a r ly s p h e r ic a l c o re .

A second c a lc u la t io n , how ever (Na 84b) (see f ig u r e 2 . 8 ) , f in d s no s t a t i c o c tu p o le shape in th e ground s ta te o f 220Ra, b u t in s te a d a p ro

nounced s o f tn e s s to w a rd n o n -z e ro 0 v a lu e s ( i . e . to w a rd o c tu p o le v ib r a t io n s ) . T h is i s su p p o rte d by th e a n a ly s is o f th e 220Ra le v e l spectrum in (Na 8 5 ) . In t h i s a n a ly s is , th e r e la t io n s h ip o f th e sp ac ing s o f le v e ls i n th e n e g a t iv e p a r i t y band to th e c o rre s p o n d in g sp ac ing s i n the p o s i t iv e p a r i t y band i s used to sug g est a t r a n s i t i o n fro m an o c tu p o le v ib r a t io n a l c h a ra c te r a t low s p in s to a s t a t i c o c tu p o le shape by s p in 14R.

The sm ooth in c re a s e in th e moment o f i n e r t i a o f 220Ra w i th in c re a s in g s p in (se e f ig u r e 5 .2 ) can be u n d e rs to o d in te rm s o f an o c tu p o le c ra n k in g model to be a s o f t n e u tro n a lig n m e n t s im i la r to th a t fo u n d in 222Th (Na 8 5 ) . T h is e x p la n a t io n , how ever, r e l i e s on a s t a t i c o c tu p o le shape w h ic h , a c c o rd in g to th e a n a ly s is m e n tio n e d above, i s p re s e n t o n ly a t h ig h e r s p in s .

5 . 5 A C O M P A R I S O N O F T H E Z = 6 0 - 6 6 I S O T O P E S W I T H T H E Z = 8 8 - 9 0

O N E S

S e v e ra l in v e s t ig a to r s have n o te d s i m i l a r i t i e s betw een th e n u c le a r sp e c tro sc o p y o f th e Ra-Th ( a c t in id e ) re g io n and t h a t o f th e Nd-Sm-Gd-Dy ( la n th n id e ) re g io n . In p a r t ic u la r , th e se s im i l a r i t i e s e x is t in th e s iz e s o f reduced a lp h a p a r t ic le decay w id th s , th e s t r u c tu r e o f le v e l s p e c tra and th e enhancement o f E l t r a n s i t io n s o v e r s in g le p a r t ic le v a l

ues (Ro 83 , Wa 83 , Go 8 5 ) .

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I n t h i s s e c t io n , we p re s e n t an e x te n s iv e com parison betw een the a c t in id e and la n th a n id e re g io n s . F ig u re s 5 .1 1 -5 .1 9 i l l u s t r a t e th e e x c it a t io n s p e c tra o f 144-152Sm and 146_152Gd as ob served in fu s io n -e v a p o ra t io n re a c t io n s ind uced by a lp h a p a r t ic le s and heavy io n s . The ground s ta te and c o rre s p o n d in g n e g a tiv e p a r i t y bands o f th e se n u c le i a re shown in f ig u r e s 5 .2 0 -5 .2 1 . As m en tioned in c h a p te r tw o , th e re i s s u b s ta n t ia l ev id ence t h a t th e se n e g a tiv e p a r i t y s ta te s a r is e p r im a r i ly fro m th e c o u p lin g o f an o c tu p o le phonon to th e g round s ta te band. C o rre sp o n d in g s p e c tra o f Ra is o to p e s o f mass 214-222 a re shown in f ig u r e 5 .2 2 . In a l l th re e is o to p ic c h a in s , a l t e r n a t in g p a r i t y sequences appear when th e re a re fo u r o r more v a le n c e n e u tro n s .

Ground s ta te reduced a lp h a p a r t ic le decay w id th s f o r Nd, Sm, Gd, Dy, Ra and Th a re shown in f ig u r e 5 .2 3 . The s c a t te r o f th e Nd and Sm v a lu e s may r e f l e c t th e d i f f i c u l t y o f m e a su rin g lo n g a lp h a p a r t ic le decay l i f e t im e s (n e a r 1 0 15 y e a r s ) . However, th e Gd and Dy h a l f - l i v e s , w h ich range fro m 7 m in u te s to 10 14 y e a rs , y ie ld reduced w id th s w h ic h a re comp a ra b le to th o se i n Ra and Th .

F ig u re 5 .2 4 compares th e v a lu e s o f B (E 1 : J -» J -1 ) /B (E 2 : J-»J-2) averaged o v e r members o f th e g round s ta te and o c tu p o le bands (see f ig u r e s 5 .2 0 -2 1 ) ; o n ly d e e x c ita t io n s to o th e r members o f th e same bands a re cons id e re d . T h is c h o ic e co rresp ond s to th e d a ta w h ic h a re a v a i la b le f o r th e Ra and Th is o to p e s . In a lm o s t a l l cases , th e gamma ra y b ra n c h in g r a t io s f o r th e la n th a n id e n u c le i shown were ta k e n fro m th e c o m p ile d v a l ues i n N u c le a r D ata S h e e ts . The e x c e p tio n s to t h i s a re 150Sm (Su 7 7 ) , i5°Gd (Ha 77) and 152Dy (Ja 7 9 ) . These v a lu e s f o r th e Nd-Sm-Gd-Dy is o topes a re s t ro n g e r th a n those ob served in s in g le p a r t ic le t r a n s i t io n s ;

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F ig u re 5 .11 Energy le v e ls o f 144Sm b e low 4 MeV observed in (c t,2n ) re a c t io n (T u 7 9 ) .

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Figure 5.12 Energy le v e l s o f 146Sm below 4 MeV observed in (a,4n) react io n (Pe 84a). States belonging to the ground s ta te and octupole bands are displayed on the r ight s ide of the f igu re .

-126-

(I0+) 3775

F ig u re 5 .13 E nerg y le v e ls o f 1<,8Sm b e low 4 .1 1 MeV observed in l i g h t - i o n ind uced fu s io n -e v a p o ra t io n re a c t io n s (Pe 8 4 b ). The ground s ta te and o c tu - p o le bands a re d is p la y e d on th e r ig h t s id e o f th e f ig u r e .

-127-

F ig u re 5 .1 4 Energ y le v e ls o f 150Sm be low 4 MeV ob served i n (c t,4n ) re a c t io n (Su 7 7 ) . The ground s ta te and o c tu p o le bands a re d is p la y e d on th e r ig h t s id e o f th e f ig u r e .

-128-

(15)" 3915

Figure 5.15 Energy le v e ls o f 152Sm observed in (a,2n) reaction (Ba 80). The ground s ta te and octupole bands are displayed on the r ight side of the f ig u re .

-129-

Figure 5.16 Energy le v e ls o f 146Gd below 4 MeV observed in l ig h t - io n induced reactions (Pe 84a).

F ig u re 5 .1 7 Energ y le v e ls o f 148Gd b e low 4 MeV observed in (a ,4 n ) re a c t io n (Pe 8 4 b ). The ground s ta te and o c tu p o le bands a re d is p la y e d on th e r ig h t s id e o f th e f ig u r e .

-131-

(I2+) 3980

F ig u re 5 .1 8 Energ y le v e ls o f 150Gd b e low 4 .2 MeV observed in (cc,4n) r e a c t io n (Ha 7 7 ) . The ground s ta te and o c tu p o le bands a re d is p la y e d on th e r ig h t s id e o f th e f ig u r e .

41874131

-132-

33663288322029062834 (9)" 2816

(8)+ •2554 • 2392

2767

7“ 221 1

2116 e * . I9 3 § / ^

1700

' /

4+ l ? R 8 ^ \ 3 ' 1 134

' //2+ 638

0+ ' - 0

Figure 5.19 Energy levels of 152Gd below 4 MeV observed in light-ion reactions (Ba 80). The ground state and octupole bands are displayed on the right side of the figure.

-133-

(I4+) 3499

F ig u re 5 .2 0 P a r t i a l ene rg y le v e l s p e c tra o f even -even Sm is o to p e s . The s ta te s shown were chosen to c o r respond to th o se observed in Ra is o to p e s . G e n e ra l ly , th e se s ta te s a re th e g round s ta te and o c tu p o le bands. The d a ta a re ta ke n from re fe re n c e s (Su 77 , Ro 82b, K i 78 , Ko 82 , Ha 79b, Tu 7 9 ) .

(15)“ 3915

6+ 2323

(I4+) 3676

9* 2807

2129

1594

1162

146c™6 2 84

4+ 774

2* 550 2+ 334

0+ 0 0+ 0148 Q 62 86

15062

(I3~)2833

2327

(9") 1879

(7“) 1506

(5“) 12223" 1041

963

52 c ™ 62® 90

-134-

F ig u re 5 .2 1 P a r t i a l ene rg y le v e l s p e c tra o f even -even Gd is o to p e s . The s ta te s shown were chosen i n a way id e n t ic a l to t h a t used f o r f ig u r e 5 .2 0 . The d a ta a re ta k e n fro m r e f e r ences (Pe 84a , Lu 84 , Zo 75 , Ha 1 1 ) .

»

(I4+) 3499

2+ 1971

8 + 2693 9“ 2 9 5 (8 *) 2767

(13") 3338

28152564

2082

1273

148Gd,64 84

23311880

(5")14713" 1123

0 + ' 0

64 8 6 64 8 8

-135-

Figure 5.22 The system atic behavior o f ex c ited s ta te s o f even-even Ra iso top es showing the a ltern atin g pari t y s tructure. The references are the same as those used in figure 5 .1 .

(

I2+ 3 2 5 3 I4+ 3 2 9 3 I6+ 3 2 8 6

0 +

I0+ 2 9 4 2 \

8 + 2 0 7 1 7

8 +V

___/ 1 8 6 46 ' 18174 +— — 1 6 3 7

2* 1381

26 8 1 i!3” 2 6 7 9

2 l 4 p n 8 8 126

11‘ 2 3 3 5

I0+ 2026>

8 +

6 +

2 +

0 +

1711

1 5 0 8

1 6 4

6 8 8

216 p n 8 8 128

17" 3390

(2D 3623

3 1 4 3

2 6 8 8

2 2 6 0

1862

1 4 9 4

1162

8 7 2

6 3 3(3~) 4 7 4

I" 4 1 2 4 + 3012++ 0 ^

3 ~ 3 1 7

V 2 4 2

2,8 Ra 8 8 1 3 02 2 0 p n

8 8 1322 2 2 Ra8 8 1 3 4

-136-

F ig u re 5 .2 3 Reduced a lp h a p a r t ic le decay w id th s fro m th e c o m p ila t io n o f (Ro 83) f o r e ven -even Nd, Sm, Gd, Dy, Ra and Th is o to p e s . The p o in t f o r 218Ra has been updated to accoun t f o r th e r e s u l t s o f (Ra 86) .

F i g u r e 5 . 2 4 A v e r a g e v a l u e s o f

B ( E 1 : J - » J - 1 ) / B ( E 2 : J - » J - 2 ) f o r y r a s t

p o s i t i v e p a r i t y s t a t e s a n d s t a t e s

f r o m t h e f i r s t e x c i t e d n e g a t i v e p a r

i t y b a n d s i n e v e n - e v e n N d , S m , G d ,

D y , R a a n d T h i s o t o p e s . T h e d a t a i s

t a k e n f r o m r e f e r e n c e s ( S u 7 7 , H a 7 7 ,

J a 7 9 , T u 7 9 , P e 8 4 a , P e 8 4 b , B a 8 0 ,

H a 7 9 a ) a s w e l l a s t h o s e u s e d f o r

f i g u r e s 5 . 5 a n d 5 . 6 .

B(EI

)/B

(E2

) (W

.u.)

-13

8-

F i g u r e 5 . 2 5 A n u m b e r o f a v e r a g e

B ( E 1 ) / B ( E 2 ) r a t i o s c h o s e n t o i l l u s

t r a t e t h e p o s s i b l e r a n g e o f v a l u e s

( H a 8 5 , D r 7 8 , D r 8 2 , L e 8 2 b ) . T h e

l o w e s t l y i n g n e g a t i v e p a r i t y s t a t e s

i n 8 2 S r a n d 1 7 6 0 s a r e b e l i e v e d t o

o r i g i n a t e p r i m a r i l y i n q u a s i p a r t i c l e

c o n f i g u r a t i o n s , w h i l e t h e n e g a t i v e

p a r i t y s t a t e s f r o m w h i c h t h e 1 7 4 W

v a l u e i s t a k e n a r e b e l i e v e d t o a r i s e

f r o m a n o c t u p o l e p h o n o n c o u p l e d t o a

q u a d r u p o l e d e f o r m e d c o r e .

B(

EI

} I

—I

-l

)/

B(

E2

i

I—

I -

2)

-139-

h o w e v e r , t h e y s t i l l f a l l a f a c t o r o f t e n b e l o w t h e l a r g e s t m e a s u r e d i n

R a a n d T h . A p p r o x i m a t e E l s t r e n g t h s c a n b e i n f e r r e d f r o m t h e b e h a v i o r

o f t h e B ( E 1 ) / B ( E 2 ) r a t i o s . F o r e x a m p l e , i n 1 4 8 S m t h e B ( E 1 ) / B ( E 2 ) a v e r

a g e i s 1 . 6 x 1 0 " 5 ( i n W . u . ) a n d t h e m e a s u r e d B ( E 2 : 2 + - » 0 + ) v a l u e i s 3 2 W . u .

( E n 8 1 ) . W e c a n e s t i m a t e t h e E l s t r e n g t h b y c a l c u l a t i n g

{ B ( E 1 ) / B ( E 2 ) } x B ( E 2 : 2 + + 0 + ) = 1 . 6 x l 0 " 5 x ( 3 2 W . u . )9 V 6

= 5 . 1 x l 0 - 4 W . u .

w h i c h i s t h u s e n h a n c e d o v e r s i n g l e p a r t i c l e v a l u e s . T h e r e l a t i v e l y

l a r g e s t r e n t h o f B ( E 1 ) / B ( E 2 ) v a l u e s i n t h e t r a n s i t i o n a l l a n t h a n i d e n u c

l e i i s e m p h a s i z e d i n f i g u r e 5 . 2 5 , w h i c h i l l u s t r a t e s t h e r a n g e o f v a l u e s

t h a t t h i s r a t i o a s s u m e s , n o t o n l y i n n u c l e i i n w h i c h o c t u p o l e p h o n o n s

a r e c o u p l e d t o a n e a r l y s p h e r i c a l c o r e , b u t a l s o w h e n a n o c t u p o l e p h o n o n

i s c o u p l e d t o a d e f o r m e d c o r e o r w h e n a q u a s i p a r t i c l e c o n f i g u r a t i o n i s

r e s p o n s i b l e f o r n e g a t i v e p a r i t y s t a t e s .

T h i s c o m p a r i s o n , w h e n t a k e n t o g e t h e r w i t h t h e a n a l y s i s o f a l a r g e r

r e g i o n i n t h e n e x t s e c t i o n , s u g g e s t s t h a t o c t u p o l e v i b r a t i o n a l b e h a v i o r

c a n n o t b e e x c l u d e d a s a p o s s i b l e o r i g i n o f t h e n e g a t i v e p a r i t y s t a t e s

a n d e n h a n c e d E l t r a n s i t i o n s o b s e r v e d i n t h e n e i g h b o r h o o d o f 2 2 0 R a ; t h i s

i s , i n t h e a b s e n c e o f f u r t h e r e v i d e n c e , a c o m p e t i n g e x p l a n a t i o n f o r t h e

a l p h a p a r t i c l e c l u s t e r i n g a n d s t a t i c o c t u p o l e d e f o r m a t i o n o n e s . W e

w o u l d a l s o n o t e t h a t , e q u i v a l e n t l y , t h e e x p e r i m e n t a l e v i d e n c e u s e d t o

s u p p o r t t h e o c t u p o l e v i b r a t i o n a l i n t e r p r e t a t i o n o f b e h a v i o r i n t h e

Z = 6 0 - 6 6 i s o t o p e s d o e s n o t e x c l u d e t h e h y p o t h e s i s t h a t a l p h a p a r t i c l e

c l u s t e r c o n f i g u r a t i o n s a r e p a r t i a l l y r e s p o n s i b l e f o r t h e o b s e r v e d p h e

-140-

nomena. However, the observation of two octupole phonon states in

147-14 8 Q ( j ( k 1 8 2 , L u 8 4 ) p r o v i d e s s t r o n g e v i d e n c e t h a t l o w - l y i n g

n e g a t i v e p a r i t y s t a t e s i n t h e t r a n s i t i o n a l l a n t h a n i d e s a r e d o m i n a t e d b y

o c t u p o l e v i b r a t i o n a l c o n f i g u r a t i o n s . T h e p o s s i b i l i t y t h a t o c t u p o l e

v i b r a t i o n a n d a l p h a p a r t i c l e c l u s t e r c o n f i g u r a t i o n s c o e x i s t i n t h e l a n

t h a n i d e r e g i o n w a s f i r s t r a i s e d i n ( l a 8 5 ) ; t h i s s u g g e s t s t h e p o s s i b i l

i t y o f s i m i l a r c o e x i s t e n c e i n t h e a c t i n i d e r e g i o n .

W e m e n t i o n e d , i n c h a p t e r t w o , t h e n o n o b s e r v a t i o n o f t w o o c t u p o l e

p h o n o n s t a t e s i n s t u d i e s o f 2 2 2 - 2 2 6 R a a n d 2 2 6 - 2 2 8 x h . N o s u c h s t a t e s

h a v e b e e n o b s e r v e d i n l i g h t e r R a a n d T h i s o t o p e s , e i t h e r ; h o w e v e r , n o

e x h a u s t i v e s e a r c h e s s u c h a s t h a t o f ( K u 7 6 ) h a v e b e e n c o n d u c t e d f o r

t h e s e n u c l e i . F u r t h e r m o r e , t h e ( H I , x n ) r e a c t i o n s m o s t c o m m o n l y u s e d t o

s t u d y t h e n e u t r o n - d e f i c i e n t l i g h t a c t i n i d e s d o n o t s i g n i f i c a n t l y p o p u

l a t e s t a t e s f a r f r o m t h e y r a s t l i n e . T h e ( 3 H e , x n ) a n d ( a , x n ) r e a c t i o n s

u s e d t o s t u d y t h e t w o o c t u p o l e p h o n o n s t a t e s i n 1 4 7 _ 1 4 8 G d c a n n o t b e u s e d

i n o u r r e g i o n b e c a u s e o f t h e l a c k o f t a r g e t m a t e r i a l s . C o n s e q u e n t l y ,

s e a r c h e s f o r t h e s e s t a t e s w i l l b e q u i t e d i f f i c u l t , m o s t p r o b a b l y r e q u i r

i n g v e r y s e n s i t i v e s t u d i e s o f a l p h a p a r t i c l e d e c a y p r o c e s s e s .

5 . 6 A C O M P A R I S O N O F 1 " A N D 3 " S T A T E S I N T H E R E G I O N S

5 6 < Z < 8 2 A N D Z > 8 2

I n t h e p r e v i o u s s e c t i o n , w e c o m p a r e d t w o r a t h e r l i m i t e d r e g i o n s o f

t h e p e r i o d i c t a b l e i n d e t a i l . W e n o w e x a m i n e a c o n s i d e r a b l y l a r g e r

d o m a i n o f t h e p e r i o d i c t a b l e , r a n g i n g f r o m Z = 5 6 ( B a r i u m ) t o Z = 9 4 ( P l u t o

n i u m ) . O u r o b j e c t i v e i s t o a n a l y z e t h e r e l a t i o n s h i p o f t h e s y s t e m a t i c

-141-

b e h a v i o r o f l o w - l y i n g n e g a t i v e p a r i t y s t a t e s i n R a a n d T h w i t h t h a t o f

s t a t e s i n P b ( w h i c h a r e c l e a r l y o f o c t u p o l e v i b r a t i o n a l o r i g i n ) , P o , R n ,

U a n d P u ( i n w h i c h t h e y m a y b e o f m i x e d c h a r a c t e r ) . F u r t h e r m o r e , w e

c o m p a r e t h e g r o s s f e a t u r e s o f t h i s s y s t e m a t i c b e h a v i o r i n Z > 8 2 n u c l e i t o

t h o s e i n t h e 5 6 < Z < 8 2 r e g i o n ( i n w h i c h t h e l o w - l y i n g n e g a t i v e p a r i t y

s t a t e s h a v e t r a d i t i o n a l l y b e e n i n t e r p r e t e d a s o c t u p o l e v i b r a t i o n a l c o n

f i g u r a t i o n s ) . A q u a l i t a t i v e a n a l y s i s o f t h i s i n f o r m a t i o n i n t e r m s o f

t h e e x p e c t e d d e p e n d e n c e o f o c t u p o l e v i b r a t i o n a l s t a t e s o n n u c l e o n n u m b e r

a n d c o r e d e f o r m a t i o n w i l l a l s o b e g i v e n i n o r d e r t o a s s e s s t h e a p p l i c a

b i l i t y o f t h i s m o d e l .

A f u r t h e r c o m m e n t c o n c e r n i n g o u r l e v e l o f u n d e r s t a n d i n g o f t h e

n a t u r e o f t h e n e g a t i v e p a r i t y s t a t e s i n t h e s e r e g i o n s i s i n o r d e r . W e

h a v e a l r e a d y m e n t i o n e d t h e p o s s i b i l i t y o f m i x e d c h a r a c t e r i n t h e t r a n

s i t i o n a l N d - S m - G d - D y a n d R a - T h r e g i o n s , a n d t h e c l e a r p o s s i b i l i t y o f

s t a t i c o c t u p o l e d e f o r m a t i o n o r a l p h a p a r t i c l e c l u s t e r i n g i n R a a n d T h .

T h e h y p o t h e s i s o f m i x e d c h a r a c t e r m u s t a l s o b e c o n s i d e r e d f o r a l m o s t a l l

o t h e r n u c l e i t h r o u g h o u t t h e Z = 5 6 - 9 4 r e g i o n ( e x c e p t f o r 2 0 8 P b a n d s e v e r a l

o f i t s n e a r e s t n e i g h b o r s ) ; t h e p r e s e n c e o f a s t a t i c o c t u p o l e d e f o r m a t i o n

i n t h e g r o u n d s t a t e o f B a i s o t o p e s n e a r m a s s 1 4 6 h a s a l s o b e e n s u g g e s t e d

( N a 8 5 ) . A r e c e n t m e a s u r e m e n t o f t h e 1 5 ‘, S m ( d , 6 L i ) 1 5 0 N d r e a c t i o n ( J a 8 2 )

s u g g e s t s t h a t w h i l e a l p h a p a r t i c l e c l u s t e r s t a t e s p r o b a b l y e x i s t a t

e x c i t a t i o n e n e r g i e s o f 2 M e V a n d a b o v e i n t h e d e f o r m e d n u c l e u s 1 5 0 N d ,

t h e l o w e s t - l y i n g n e g a t i v e p a r i t y s t a t e s a r e d o m i n a t e d b y a n o c t u p o l e

v i b r a t i o n a l c o n f i g u r a t i o n ( l a 8 5 ) .

W e b e g i n t h i s c o m p a r i s o n b y e x a m i n i n g t h e b e h a v i o r o f E ( 3 " ) i n t h e

t w o r e g i o n s . I n f i g u r e 5 . 2 6 , E ( 3 " ) i s p l o t t e d a g a i n s t t h e n u m b e r o f

-142-

F i g u r e 5 . 2 6 S y s t e m a t i c b e h a v i o r o f

3 j / s t a t e s i n 8 2 < N < 1 2 6 a n d N > 1 2 6

r e g i o n s . T h e a b s c i s s a i s t h e v a l e n c e

n e u t r o n n u m b e r . O p e n c i r c l e s d e n o t e

t e n t a t i v e a s s i g n m e n t s . D a t a f r o m ( A u

8 3 , B a 8 0 , B a 8 1 , B a 8 2 , B o 8 3 b , B r

8 4 , D e 7 7 , D e 8 1 , D o 7 9 , F a 8 2 , F e

8 3 , F i 8 4 , G a 8 3 b , G i 8 3 , H a 7 7 , H a

7 9 a , H e 8 2 , H e 8 5 , H i 8 6 , H o 7 7 b , K a

8 0 , L e 7 8 , L e 8 0 a , L e 8 2 b , L e 8 5 , M a

7 7 a , M a 8 0 , M e 8 1 , M o 8 2 , P e 7 9 a , P e

7 9 b , P e 8 1 , P e 8 2 , P e 8 4 a , P e 8 4 b , P e

8 4 c , S c 7 9 a , S c 8 0 b , S h 8 3 c , S i 8 1 ,

S p 7 8 , S p 8 0 , S t 7 7 , T u 7 9 , W a 8 1 , Z o

8 0 , Z u 8 2 ) a n d t h e p r e s e n t w o r k .

3 2 0 0

2 8 0 0

2 4 0 0

2000

1 6 0 0

1200

8 0 0

4 0 0

0

3 2 0 0

2 8 0 0

2 4 0 0

2000

1 6 0 0

1200

8 0 0

4 0 0

00 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4

Nn

v a l e n c e n e u t r o n s . I n b o t h r e g i o n s t h e g r o s s b e h a v i o r i s t h e s a m e :

E ( 3 " ) d e c r e a s e s a s t h e f i r s t n e u t r o n s a r e a d d e d t o t h e c l o s e d s h e l l a n d

f i n a l l y r e a c h e s a c o n s t a n t v a l u e i n d e f o r m e d i s o t o p e s . F o r t h e c a s e o f

o c t u p o l e v i b r a t i o n s , t h i s t r e n d a r i s e s f r o m t h e i m p o r t a n c e o f t h e s t r o n g

o c t u p o l e t r a n s i t i o n s b e t w e e n t w o o r b i t s w i t h i n a m a j o r s h e l l w h o s e a n g u

l a r m o m e n t u m d i f f e r s b y 3 f i : c35 / 2 _ b l l / 2 ^ o r t b e 5 0 < Z < 8 2 s h e l l ,

f y ^ 2 ~ i 13^2 f ° r b o t h p r o t o n a n d n e u t r o n 8 2 - 1 2 6 s h e l l s a n d 9 g / 2 ~ 3 i 5 / 2 ^ o r

N £ 1 2 6 ( s e e t h e N i l s s o n s h e l l m o d e l d i a g r a m s i n f i g u r e s 5 . 2 7 - 5 . 3 0 ) . W e

w i l l c a l l s u c h a p a i r o f o r b i t s a A £ = 3 p a i r . A s p a r t i c l e s a r e a d d e d t o

t h e l o w e r e n e r g y m e m b e r o f s u c h a p a i r , a d d i t i o n a l o n e p a r t i c l e - o n e h o l e

e x c i t a t i o n s o f E 3 m u l t i p o l a r i t y b e c o m e a v a i l a b l e ; c o n s e q u e n t l y , t h e

e n e r g y o f t h e c o h e r e n t s u m o f t h e s e e x c i t a t i o n s , w h i c h i s t h e c o l l e c t i v e

3 " s t a t e , i s d r i v e n d o w n i n e x c i t a t i o n e n e r g y ( R i 8 0 , B o 7 5 ) . I n t h e

l i g h t e r r e g i o n ( w h i c h w e w i l l h e n c e f o r t h c a l l t h e l a n t h a n i d e r e g i o n ) ,

t h e d e c l i n e o f E ( 3 " ) w i t h i n c r e a s i n g n e u t r o n n u m b e r i n t h e n e a r l y s p h e r

i c a l n u c l e i c l o s e t o N = 8 2 c a n b e t r a c e d t o t h e f i l l i n g o f t h e n e u “

t r o n o r b i t a l . W h e n d e f o r m a t i o n s e t s i n , t h e o r d e r i n g o f t h e s i n g l e p a r

t i c l e o r b i t a l s i s c o m p l i c a t e d u n d e r t h e N i l s s o n s c h e m e a n d t h i s s i m p l e

p i c t u r e b r e a k s d o w n .

I n t h e g r o u n d s t a t e b a n d s o f t h e l a n t h a n i d e n u c l e i , d e f o r m a t i o n

s e t s i n r a t h e r s u d d e n l y a t N = 9 0 ( C a 8 5 ) . H o w e v e r , b y c o n s i d e r i n g t h e

s y s t e m a t i c b e h a v i o r o f 3 ~ s t a t e s w e c a n s e e t h a t s u f f i c i e n t d e f o r m a t i o n

e x i s t s ( a t l e a s t i n t h e o c t u p o l e b a n d ) t o r e n d e r t h e s p h e r i c a l p i c t u r e

i n v a l i d a t a n e v e n l o w e r n e u t r o n n u m b e r . F i g u r e 5 . 3 1 s h o w s E ( 3 - ) f o r

f o u r i s o t o n i c c h a i n s p l o t t e d a s a f u n c t i o n o f v a l e n c e p r o t o n n u m b e r .

J u s t a s i n t h e n e u t r o n s h e l l , E ( 3 ~ ) w i l l d e c r e a s e i n a s p h e r i c a l n u c l e u s

-144-

F i g u r e 5 . 2 7 A N i l s s o n d i a g r a m f o r

p r o t o n s , 5 0 < Z £ 8 2 ( L e 7 8 ) . T h e n u m

b e r s o n t h e f i g u r e a r e d e f i n e d a s

f o l l o w s : E i s t h e e n e r g y o f t h e

o r b i t a l ; i s t h e o s c i l l a t o r e n e r g y

f o r t h e s h e l l m o d e l ( g i v e n a p p r o x i

m a t e l y b y 4 1 / A * ^ 3 ) ; e i s t h e

s t r e t c h e d q u a d r u p o l e d e f o r m a t i o n

p a r a m e t e r , w h i c h i s a p p r o x i m a t e l y

e q u a l t o ( R 1 8 0 ) • T ^ e l e t t e r a n d

n u m b e r s h o w n a t z e r o d e f o r m a t i o n

( i . e . g i v e 1 a n d j , r e s p e c

t i v e l y . T h e n o t a t i o n o f t h e f o r m f t [ N

n A ] g i v e s t h e N i l s s o n a s y m p t o t i c z

q u a n t u m n u m b e r s ( E i 7 5 ) ; ft i s t h e

p r o j e c t i o n o f j o n t h e z - a x i s o f t h e

i n t r i n s i c f r a m e ; N i s t h e o s c i l l a t o r

q u a n t u m n u m b e r ; n i s t h e n u m b e r o f z

o s c i l l a t o r q u a n t a i n t h e d i r e c t i o n o f

t h e z - a x i s o f t h e i n t r i n s i c f r a m e f o r

l a r g e d e f o r m a t i o n ; a n d A i s t h e p r o

j e c t i o n o f £ o n t h e z - a x i s o f t h e

i n t r i n s i c f r a m e . S o l i d a n d d a s h e d

c u r v e s d e n o t e p o s i t i v e a n d n e g a t i v e

p a r i t y o r b i t a l s , r e s p e c t i v e l y .

9*0 g-Q fr'O £‘0

(Wlt/l -

- s n -

F i g u r e 5 . 2 8 A N i l s s o n d i a g r a m f o r

n e u t r o n s , 8 2 < N £ 1 2 6 ( L e 7 8 ) .

Uf»)r.'t

liwlt.t * \ ,\ i \ \\ j[IZS]:/C--

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(it9]r/t ftttlt/i-N ,'“«{/tto2TF7rCt

Il“ , t / r i i c 5 u 7 T ^

Itttlt/* Un]i/t t Imlt/n Icrelt It ."|K9)£/1 s.ICOt)£/l*

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hc

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(€

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Figure 5.29 A Nilsson diagram for protons, Z>82 (Le 78).

9*0

llMll'C

I'tltU'i[m!:/c .Imlt t front**IlKll-t Ito»lr-t[I t f lt 'l-----|088]M

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|5H I l | If I

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E/1iaJ0 (€)

Figure 5.30 A Nilsson diagram for neutrons, N>126 (Le 78).

E/tiOJ0 (€ )

7/2(833],9/218441

,1/2(750153/21741]

/ 3/2[732]

3 /2 [402] 1/2(7411 1/2[400]

7/2(404]

1/2(871]

^ 13/2(707]

11/2(404] j / 3/2(7321

^ 3 /2 (8 3 1 ] y 7/2(943]

3 / 2 (8 3 3 1 1/2(981]9/2(404] 1/2(851]

^ 13/2(714]

1/2(9701

« s 1— ® o ® an u o> m * m— « a o O M

-8 7 1 -

Figure 5.31 The systematic behavior of first excited 3" states in the 82<N<90 and N>126 regions. The abscissa is the valence proton number. Open circles denote tentative assignments. The data are taken- from (Ba 80, Bo 83b, Ga 83b, Gi 83, Ha 77,Ha 79a, He 82, Ho 77, Le 78, Mo 82,Pe 79b, Pe 81, Pe 82, Pe 84a, Pe 84b,Pe 84c, Sc 80b, Tu 79, Zo 80) and thepresent work.

3 2 0 0

2 8 0 0

2 4 0 0

2 0 0 0

1 6 0 0

1 2 0 0

8 0 0

4 0 0

0

3 2 0 0

2 8 0 0

2 4 0 0

2 0 0 0

1 6 0 0

1 2 0 0

8 0 0

4 0 0

0

-149-

0 2 4 6 , 8 10 12 14 16 18

the one in the proton shell. Because is the lowest orbital

in the proton shell, the dc orbital does not fill until N =14. This5/2 p

simple model works quite nicely for the N=82 and N=84 chains. However,

by N=88 the nuclei are sufficiently deformed to render this scheme

invalid.

A similar analysis can qualitatively explain behavior in the Z>82

region (which we will call the actinide region). The A£=3 orbital pairs

are 9g/2”^i5/2 and ^7/2~413/2 4n tke neutron and Proton shells, respec

tively. Near N=126, the filling of the vgQ ,~ orbital reduces E(3_) withy / £increasing neutron number (This was first noted in (Sh 80)). The Rn

isotopes reflect this behavior very well. 222Rn is not very deformed -

[E(4+)/E(2+)=2.4] - but E(3") seems to reach a minimum at Nn=10.

Instead of being caused by the onset of deformation, as in the lanthan

ide region, this flattening out of E(3-) in the Rn isotopes is most

probably the result of the filling of the vgQ/0 orbital that we wouldy / £expect in a simple picture for a nearly spherical nucleus.

The situation in Ra and Th is somewhat different. The Ra and Th

curves reach minima at 222-224Ra [E(4+)/E(2+)=2.7 and 3.0, respectively]

and 224-226xh [E(4+)/E(2+)=3.0 and 3.1, respectively], in which the qua

drupole deformation is large enough to invalidate the simple single par

ticle level ordering found in spherical nuclei. If we were to assume an

octupole vibrational behavior for Ra and Th, the quadrupole deformation

would be responsible for the minima as in the lanthanides.

The proton dependence in the actinide region is rather striking.

Not only is E(3_) several hundered keV lower in Ra and Th than in Rn,

-150-

as the lower proton orbital of the A£=3 pair is filled. This orbital is

but also E(3") seems to be independent of proton number for Z>88 and

N>134. We can, though, propose a schematic explanation for this behav

ior in terms of octupole vibrations. The picture would at first seem to

be different from that of the neutron shell because of the near degener

acy of the A£=3 orbital partners in the proton shell (fy72 anc* ^13/2^ ’

However, we would expect that in a spherical nucleus E(3") would

decrease as the Trhg/2 orbital is filled and the occupation probabilities

of the irfy 2 ancl irl'i3/2 orbitals increase. Once again, this is consis

tent with the limited information we have on N<132 isotopes. The flat

tening of E(3~) vs. Np curves for N>134 Ra and Th isotopes may be caused

by the onset of quadrupole deformation, just as in the N=88 chain in the

lanthanide region.

It would be enlightening to test the spherical nucleus expectation

for the dependence of E(3") on proton number against the actual behavior

in Rn, Ra and Th isotopes of N<130, as well as in 220Th and 222Th. How

ever, the experimental difficulties are formidable. The light ion reac

tions used to study the non-yrast structure of many lanthanide nuclei

are not practical for the light actinide isotopes because of a lack of

target materials. Consequently, the most promising avenues for investi

gation are the studies of alpha particle decay chains of short-lived Th

and U isotopes and (HI,xn) reactions close to the Coulomb barrier. With

the former method, the negative parity states would be difficult to

detect at excitation energies above 1 MeV, such as those we would expect

in the N=128 isotones. The latter method presents a similar difficulty:

in the N<130 isotopes we would expect the low-spin negative parity

states to be far from the yrast line, and, thus, to be populated quite

-151-

weakly.

In chapter 2, we mentioned how the position of the 1“ state arising

from the (3" octupole phonon)®2+ configuration evolves with respect to

the (3~ octupole phonon)0O+gs state as quadrupole deformation sets in.

In an ideally spherical nucleus, E(l-)-E(3-), the difference in the

excitation energies of these two states, is nearly equal to the excita

tion energy of the first 2+ state. As deformation, E(l“)-E(3')

approaches zero and finally settles at a negative value. At the

ITdeformed limit, this 1“ state is actually the band head of the K =0“

octupole vibrational band.

The E(3")-E(l") plot against for the lanthanide region (figure

5.32) shows this trend. For N >8, the l- state has fallen below the 3"' n

state (E(3-)-E(l")>0) in almost all isotopes. However, there is some

chaos in the plot arising from the range of deformations for a given

and the Coriolis coupling between K ^ O - and K ^ l " octupole bands for

nuclei at the deformed limit.

We can make these data,appear more orderly by plotting against

E(4+)/E(2+), a quantity that reflects deformation more directly (see

figure 5.33). In this plot, the behavior of E(3“)-E(l') becomes trans

parent. The 1' state reaches the 3' in energy at E(4+)/E(2+ )=2.5 in the

lanthanides, and the maximum of 80 keV at E(4+)/E(2+)=2.7. The scatter

reflecting Coriolis coupling is evident at the deformed limit

[E(4+)/E(2+)=3.3]. This plot demonstrates the dominant role that the

quadrupole deformation plays in determining the relative spacing of

these two levels for E(4+)/E(2+)<3.0.

-152-

If we move on to the actinide region, we can see that the limited

Figure 5.32 The systematic behavior of E(31')-E(l1-) in the 56<Z^70 andZ>82 regions. The abscissa is the valence neutron number. Open circles denote tentative assignments. The data are taken from (Ba 80, Ba 82, De77, Fi 84, Gi 83, Ha 77 Ha 79a, He85, Hi 86, Ho 77b, Ka 80, Ko 76, Le78, Le 80a, Le 85, Pe 79b, Pe 84a, Pe84b, Pe 84c, Sc 80a, Wa 81, Zo 80, Zu 82, Ku 76, Ku 77, Ku 78, Pe 81, Ga83b, Bo 83b) and the present work.

(keV

) .

..

.

(ke

V)

-153-

Figure 5.33 The systematic behavior of E(31-)-E(l1-) in the 56<Z<70 andZ>82 regions. The abscissa is E(4^+)/E(2^+). Parentheses denotetentative assignments. The references are those used for figure 5.23.

UZ)

3 A

l*)3

E( 370) (Si ^o O O

E ( 3 p - E ( l p (keV)

01o ro J.o o— roo o o>o uto OJoo

ro -lo oo o o O o o

—

1 l 1 1

~ 3

I 1 1 1| 1

1 1

1 E(3p-E(lp

• Rn

ORa

■ Th —

O i

►■<cr■ <J > m o o , << a.

mOJ.

□ •4 0 * -1 EAZOQ) 'T' 3 a • 0 1 m

—” .

5 >

a

• — <■— £ -- — n i □ *• —

— • — --O, w

•

_ 0•

0 “ w —

— -- __

8* <3 ? ““

0■_ > D

——5

-- —— -- -- ——

1 1 1«

1— —

1 1 V 1^1—

-15

4-

data behave in a mariginally more coherent fashion on the E(4+)/E(2+)

plot than on the plot. The only point falling well off the line on

the E(4+ )/E(2+) plot is 218Ra [at E(4+)/E(2+)=l.9], in which a state

only tentatively observed was given a 1' assignment (Ga 83b). One

interesting feature of the actinide region plot of E(3")-E(l") against

E(4+)/E(2+) is that E(3“)-E(l_) seems to reach its 80 keV maximum at

E(4+)/E(2+)=2.3-2.5, representing a less deformed shape than the corre

sponding point in the lanthanide plot. Recalling, however, the observa

tion that the N=130 and N=132 Ra and Th isotopes display near-vibra

tional behavior in the positive parity bands but rotor-like patterns in

their negative parity bands, we may speculate that the negative parity

states have a larger quadrupole deformation than do the positive parity

states, and that the E(3')-E(l') dependence reflects this larger defor

mation.

We can take two views of this behavior in the actinide region. On

the one hand, it was noted in (Ci 85) that the onset of quadrupole

deformation occurs significantly more slowly (in terms of the addition

of protons and neutrons to the doubly magic nucleus 208Pb) than in other

regions of the periodic table with similar shell structure. We may

speculate that if octupole vibrations are indeed responsible for the

negative parity states, then the vibrations somehow negate the mechanism

causing this delay.

On the other hand, if the present tentative assignment of a 1”

state in 218Ra proves to be correct, it will be difficult to support the

octupole vibrational picture for the known negative parity states in

218Ra, in particular, and for the Ra isotopes (and possibly the Th iso

-155-

topes) in general. The first reason for this would be the gross

inconsistency between the E(3')-E(l") vs. E(4+)/E(2+) plots of the lan

thanide and actinide regions. The second reason would be the apparent

contradiction in having a 1" octupole vibrational state so far below the

corresponding 3_ state in such a spherical nucleus. The placement of

the 1~ state in 218Ra can be determined by a careful analysis of the

alpha particle decay of 222Th. Such an experiment is in progress in

this Laboratory.

To summarize, although we have not presented evidence that can dis

criminate between the octupole and alpha particle cluster shapes, we

have been able to demonstrate that the presently available information

is consistent with an octupole vibrational interpretation. If we con

sider a static octupole deformation to be an extreme case of an "octu

pole collectivity", as in the quadrupole case, then our analysis is also

consistent with the static shape. Further, we have suggested experimen

tal tests which may prove capable of discriminating between these

shapes. We may conclude that the resolution of these questions depends

on the search for more experimental data, most importantly on low-spin

states located off the yrast line.

5 . 7 S Y S T E M A T I C B E H A V I O R O F O D D - A N U C L E I N E A R A = 2 1 9

The interpretation of the energy level spectrum of a heavy odd-A

nucleus usually begins with an examination of the spectra of neighboring

even-even nuclei. We also begin in this way. The A<220 Ra and Th iso-

-156-

topes are generally spherical and nearly spherical, as shown by

E(4+)/E(2+) ratios (1.69 for 216Rn, 1.90 for 218Ra, 2.30 for 220Ra, 1.73

for 218Th and 2.04 for 220Th). Under these conditions, we would expect

that a weak coupling-like model, possibly including second order

effects, could account for the spectroscopy of odd-A neighbors. This

prediction is supported by the ground state spin-parities of 126<N<131,

82<Z£89 odd-A nuclei. Of the four odd- neutron and eight odd-proton

nuclei in this group whose ground states have been studied (Le 78, De

83a, De 85 and Co 85), each is reproduced by the weak coupling pre

dictions .

A calculation for 219Ra using an octupole Nilsson-like strong

coupling model has predicted a ground state spin of l/2+ (Na 85). The

underlying premise for this calculation is that 219Ra has a static

ground state octupole deformation, a view which is in conflict with

potential energy calculations of even-even neighbors by the same author

(Na 84b, Na 85).

We now examine the spectroscopy of four odd-A nuclei in this

region, 217~2*9Ac and 217"219Ra. Figure 5.34 shows how these isotopes

conform to the general pattern associated with weak coupling. In this

model, each state of the core generates a multiplet of states having

spins |jSp"Jcl-J^jSp+Jc' which only the one or two states of highest

spin would be observed in heavy ion reactions, which to date have been

used to study each of these isotopes. If the state of spin J =j„+Jmax sp c

falls below the J -1 state in a given multiplet, only the J state max 3 maxwill be observed. Both J and J -1 states are seen if the reversemax max

1 5 7 -

is true.

Figure 5.34 Particle-core coupling behavior for A=217 and 219. Lines connect corresponding weak coupling states. Single particle configurations are indicated under each odd-A band. The data is taken from (De 85, Bo 85, Ch 82, It 83, Dr 85, Ro 84, Ga 83a) and the present work.

keV

3400

3200

3000

2800

2600

2400

220020001800

1600

1400

12001000800

600

400

200

- !5/2~17/2"

11/2” - 13/r"

9/2'irhc'9/2

PARTICLE CORE COUPLING

4+

2*

0*

2189 0 Th

10*ESTIMATED B(E2)=20 W.u.

B ( E I ) « 7 3 xI0*4 W.u.

0* 9/2* 11/2* 15/2”•V9/2 vln/2 riis/a

2 l 6 o - ?I7C88Ra 88 Ra

The even-even core nuclei in figure 5.34 were chosen to illustrate

a particular model which we discuss below. From a more general view

point, it would also be appropriate to illustrate as a core nucleus the

other adjacent even-even neighbor of each odd-A isotope (216Ra for

217Ac, 218Ra for 217Ra, 218Ra for 219Ac, and 220Ra for 219Ra).

We find one exception to the weak coupling pattern. The band head

ITof the negative parity band in 219Ra is tentatively assigned to have J

= (11/2'); but the unique parity orbital in the N>126 neutron shell is

the j 15/2 one‘ Clearly, this state has an origin different from the

classical weak coupling of a j 1T= 15/2“ particle to the 0+ ground state of

the even-even core. Such a configuration of levels, the "j-2 anomaly"

mentioned in chapter 2, signals that additional strength in the core

particle interaction will invalidate the weak coupling picture. The

"j-2 anomaly" has been observed and treated theoretically in 75Se (To

84) and i»3Tc (De 83b).

The core-particle interaction, which is dominated by the 0 -qc core sp

term near A=220, can be strengthened in two ways. The first is to

increase the single particle quadrupole moment, which is given by (Br

77)

Qsp = b(N+3/2) (2jsp-l)/(2jsp+2) (5.6.1)

where b is a constant and N is the major oscillator shell of the single

particle. We can explain the apparent difference in coupling strengths

between the two observed bands with (5.6.1). For gg^ and ^15/2 orkx~

tals, this expression gives

-159-

V 99/2> = 5‘5b

-160-

The extra strength that the j,c/„ orbital lends to the Q ’O prod- * J15/2 *core xsp ruct seems to be sufficient in 2'9Ra to trigger the appearance of the j-2

anomaly.

An increase in core deformation also increases the interaction. We

would expect, for example, that the core of 22'Ra would be more deformed

than that of 219Ra. In turn, the j-2 anomaly might occur in the vgg 2 band. The ground state spin of 5/2+ measured in 22'Ra (Ah 83) could

then be explained in terms of intermediate coupling to a nearly spheri

cal core instead of by strong coupling to a statically deformed core as

in (Na 85). This suggestion is interesting because it is not clear

whether 22'Ra has a significant static deformation of any multipolarity.

An interesting suggestion regarding the nature of weak coupling

bands in the isotopes shown in figure 5.34 has been made (Co 86) to

account for the markedly different level spectra of the isobars 219Ra

and 2l9Ac. As noted in chapter 2, a prolate core has

and

V j15/2> = 7b‘

< J |1 Y<2> || J > < 0,core core

and the single particle reduced matrix element is

< j || y <2> || jgp > > 0 for a particle, and

< 0 for a hole.

A particle coupled to a slightly prolate core yields weak coupling

multiple ts in which the J state is energetically below the J -1max 3 2 maxstate. The opposite would be true for a weakly coupled hole.

In the case of the common K=0 collective band with the spin

ITsequence J =0+,2+,4+, etc., and no negative parity states, this coupling

results in a simple rule. When a particle is coupled to a slightly pro

late core, there is one yrast state in each weak coupling multiplet; for

a hole, there are two yrast states in each multiplet. When the band of

core states has an alternating parity structure, as in our case, this

rule is no longer strictly true. We can, however, state that, in a

(HI,xn) reaction, it should be considerably easier to observe two states

from each weak coupling multiplet when a hole is coupled to a slightly

prolate core than when a particle is involved.

The abscence of J -1 states from the observed structure of the maxground state vgg/,2 band in 219Ra can be explained simply as the result

of a particle coupling to the slightly prolate 218Ra core. On the other

hand, both J and J -1 states of the weak coupling multiplets of max max219 Ac are observed. This can be described in the simple coupling pic

ture as a nhg^ bole coupled to a 220Th core. This picture is consis

tent not only with 219Ac and 219Ra (the odd neutron has a particle

nature because the vgg^2 orbital is only half full), but also with the

observed spectra of 217Ac and 217Ra.

In the abscence of the pairing force between protons coupled to

angular momentum zero, it is clear that the seven protons above the

magic core in Ac would more than half fill the bg^ orbital, which is

the lowest in this proton shell, and cause the hole behavior that we

seem to observe. However, the pairing force acts to smear out nucleons

-161-

over several orbitals. The effect of the pairing force is calculated

using a BCS theory which is essentially identical to that used for

superconductors (Si 65). The parameters needed for such a calculation

are the energies of the single particle levels and A, the strength of

the pairing interaction; the output includes the occupation numbers for

the individual orbitals. In general, the occupation probabilities are

smeared out over a range of orbitals falling within an energy A of the

orbital which would be the last filled in the absence of pairing.

The importance of pairing in weak particle-core coupling is empha

sized by the model presented in (Ha 75). A factor arising from the

pairing force and appearing as the occupation number of the single par

ticle orbital is included in the core-particle interaction. In short,

if the orbital is less than 50% occupied, then the weak coupling multi

plet is ordered just as in the case of particle coupling; for occupation

of greater than 50%, the multiplet is ordered as it would be for a hole.

In order for the h g ^ Pr°ton orbital to have an occupation number

of greater than 50% in Ac, where only seven valence protons are present,

there must be a separation between the hg^2 orbital and the next highest

orbital ( i - ^ / Z ' aS seen 5.29) of an amount nearly equal to A>

In (Bo 69), a simple estimate of A is given, namely:

A = 12 / A1/2 (MeV).

For A=220, this expression yields A=810 keV. From figure 5.29, we can

estimate the gap between the h g ^ and •<1 3/2 orbitals to be 800 keV at

f$2=0. We conclude, therefore, that our interpretation of 219Ac in terms

of a proton hole coupled to 220Th has a reasonable basis. It would be

-162-

interesting to perform a careful BCS calculation to investigate this

question further.

-1 6 3 -

6 S U M M A R Y A N D C O N C L U S I O N S

We have measured yrast high spin states in 220Ra and 216Rn using

techniques of gamma-ray spectroscopy. In 220Ra, we observed the alter

nating parity behavior at high spins that is characteristic of Ra and Th

isotopes near mass 220 and a signature for reflection asymmetric intrin

sic nuclear shape. Further, we have measured B(E1:J-»J-1)/B(E2:J-»J-2)

ratios in this nucleus. Using estimates for B(E2) values based on an

expression that accounts for the systematic behavior of these values (Bo

85), we have mapped the evolution of El matrix elements of Ra isotopes

as a function of N. This information, together with data on Th iso

topes, supports the view that B(E1) values vary smoothly as a function

of both N and Z in even-even Ra and Th nuclei.

The smooth variation of B(E1)/B(E2) values is correlated to trends

in both ground state alpha particle decay reduced widths and alpha par

ticle decay hindrance factors for 1" and 3" states. This correlation is

evidence that all three trends are signatures of a single nuclear phe

nomenon.

As in previous measurements on 216Ra and 218Th, the yrast spectrum

of 216Rn displays no evidence for negative parity states below spin llh.

This result emphasizes that the mechanisms leading to intrinsic reflec

tion asymmetry in this region depend upon Z as well as upon N.

We then used these variations, as well as several observed similar

ities between the actinide and lanthanide regions, to search for

insights regarding the nature of reflection asymmetry in Ra and Th.

Even though there does not appear to be any existing data that can dis-

criminate unambiguously among alpha particle clustering, octupole vibra

tion and static octupole deformation models, our comparison of the lan

thanide and actinide regions leads us to suggest several experimental

approaches to resolving this question. These include searches for two

octupole phonon states as well as for low-lying 1" and 3" states in sev

eral neutron deficient Ra and Th isotopes. In general, more data are

needed to allow a clear decision about whether one of these mechanisms

is dominant or whether they coexist.

From a theoretical viewpoint, it is important to investigate

whether an octupole shape can account for the spectroscopic trends men

tioned above. In particular, it must be determined whether an octupole

shaped nucleus can inherently possess a large alpha particle decay

width. The role of shell structure in these observables should also be

examined.

Our measurement of the ground state band and a side band of 219Ra

by gamma-ray spectroscopic techniques is entirely consistent with a weak

coupling interpretation. However, the appearance of the j-2 anomaly in

the vjj 2 s4c*e band indicates that we are on the boundary of the region

in which weak coupling is an appropriate approach.

In conclusion, although we have obtained substantial new evidence

for reflection asymmetric nuclear shapes in the light actinides, the

present body of systematic data is not sufficient to distinguish among

the alpha particle cluster, octupole vibration and static octupole

deformation models because of similarities in the models and experimen

tal difficulties in observing low-spin, non-yrast states in this mass

region. Enough of this information is now available, however, to sug

-165-

gest specific directions for future experimental work to resolve these

questions. In addition, odd-A nuclei in this immediate vicinity behave

in a way very similar to nuclei of equivalent quadrupole deformation in

other regions of the periodic table. The latter finding suggests that

our understanding of such behavior in reflection symmetric systems will

apply, in at least a crude extrapolation, to the reflection asymmetric

shapes in transitional actinide nuclei.

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